1
vote
4answers
33 views

Simplifying/solving a logarithm $\log_24^{2n}$

Need help with simplifying this logarithm. $$\log_24^{2n}$$ Would I just pull the 2n to the front: $$2n*\log_24$$ So it would simplify to $$4n$$ Is this correct or am I completely wrong?
4
votes
2answers
20 views

Condensing Fractional Logarithms

Does the following condense to the following: $\log_2z+(\log_2x)/2+(\log_2y)/2 = \log_2(z\sqrt{x}\sqrt{y})$ or to $\log_2(z\sqrt{xy})$ ?
1
vote
2answers
38 views

Proof of logarithmic identity $\log_g x=\log_a x\cdot\log_g a$

I have to prove the alleged link between the logarithms in base g and a $$\log_g x=\log_a x\cdot\log_g a$$ I know that this can be written as: $$\frac{\ln x}{\ln g}=\frac{\ln x}{\ln g}$$ But does ...
3
votes
2answers
89 views

How to express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k=\log_2 (\sqrt{9} + \sqrt{5})$?

If $$k=\log_2 (\sqrt{9} + \sqrt{5})$$ express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k$.
3
votes
2answers
80 views

What is the meaning of this Wolfram Alpha result when calculating $3^p = 4^q$?

I would like to know are the some $p \in \mathbb{N}$ and $q \in\mathbb{N}$ for $3^p = 4^q$ except the trivial $p = q = 0$. So, I entered the expression into Wolfram Alpha, which returned the result ...
1
vote
2answers
102 views

Can one use logarithms to solve the equations $2=3^x + x$ and $2=3^x x$?

Could someone explain how would you solve: $$2=3^x + x$$ and $$2=3^x \cdot x$$ I can only solve halfway through. And why is $$10^{\log (x)}= x$$ Thanks
5
votes
5answers
55 views

Pre-calculus algebra logarithm question

I don't understand how to solve this equation. Been struggling with it and don't know how to start: $$\log_2x=8+9\log_x2$$ Can someone please help me out?
7
votes
3answers
173 views

Logarithm Equality

$$\sqrt{\log_x\left(\sqrt{3x}\right)} \cdot \log_3 x = -1$$ I am not entirely sure how to go about solving for $x$. I cannot square each side because the product isn't $≥ 0$, I can't think of any ...
2
votes
2answers
36 views

Expansion of Logarithms with Cube Roots

Does the following expand to the following $$ \log_6(11^6\sqrt[3]{12}) $$ = $ 6\log_6(11) + \log_6 (\sqrt[3]{12})$
1
vote
5answers
112 views

Identity with logarithms?

Is it correct? $$(\log\,n)^{(\log\,n)} = n^ {(\log\,\log\,n)} $$ If yes and they are equal, how can I get $(\log n)^{\log n}$ from $n^{\log \log n}$ ? Thanks.
0
votes
2answers
37 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
-1
votes
2answers
57 views

Is $f(x) = 2 + \ln x$ another way to write $f(x) =\log_e x +2$?

I just want to make sure I am correctly understaning this concept. $f(x) = 2 + \ln x$ is the same as $f(x) =\log_e x +2$ Thus my T graph would look like so: e^y|x+2 -3|2.049 -2|2.135 ...
2
votes
2answers
52 views

Does loga/logb = log(a^(1/logb))?

I know $\log(a^b)=b\log(a)$. However, Wolfram Alpha tells me that $\frac{\log(a)}{\log(b)}$ does not equal $\log(a^\frac{1}{\log(b)})$. Is Wolfram Alpha correct? If it is, why is it correct? I'm ...
0
votes
1answer
15 views

What is the logaritmic form of $v=Ae^{Bi}$

I am reading a scientific paper, which uses a model of the form $v=Ae^{Bi}$ and then it says that this model has the following logarithmic form $\ln (v) = Bi + ln(A)$ where A is a constant. But the ...
1
vote
3answers
57 views

How to use the logarithm method to solve $18^{4x-3}=(54\sqrt{2})^{3x-4}$ for $x$?

What value will satisfy this equation: $$18^{4x-3}=(54\sqrt{2})^{3x-4}$$ Please use the logarithm method. I am having a problem in expressing $54\sqrt{2}$ in the power of $18$. My book simply ...
3
votes
4answers
75 views

Exponential equation: $2e^{-x} - e^{-2x}=0.$ [closed]

$2e^{-x} - e^{-2x}=0.$ the correct answer is $x=-\ln2$. How do I get there?
5
votes
3answers
118 views

Solve for $x$ in the equation [closed]

Please help me to solve for x using maybe logarithm or exponential rules (or both) $$ 5^x=2 \cdot 3^x $$
1
vote
4answers
116 views

Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$

I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers.
3
votes
2answers
70 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
0
votes
1answer
70 views

Solving the system with logarithms

I tried solving the system $ \begin{cases} (4x)^{\log_2 (2y)} = 64 \\ (8y)^{\log_2 (2y)} = 256 \end{cases} $ several times but still keep getting wrong solutions.
3
votes
4answers
101 views

Solving $e^{4x}+3e^{2x}-28=0$

How to solve this equation: $$e^{4x}+3e^{2x}-28=0$$ I don't know how to solve this problem. I read over another example, $e^{2x}-2e^x-8=0,$ and it said that $e^{2x}$ is $e$ to the $x$ squared, ...
1
vote
2answers
61 views

Having trouble solving $\log (x − 21) = 2 − \log x$ for $x$

I'm having trouble with this problem: $\log (x − 21) = 2 − \log x$, solve for $x$. I'm coming up with $x=-5$ but that can't be right.
1
vote
2answers
67 views

Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
0
votes
2answers
62 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
2
votes
2answers
147 views

Solve for $x$: $\frac1e = e^{2x}$

I tried making it to $e^{-1} = e^{2x}$ and had the exponents equal each other $-1=2x$ and the I solved for $x$, making it $x=-1/2$, but that answer is wrong. please help I don't know why that ...
1
vote
4answers
1k views

What's wrong with my aproach to solving this equation with multiple logarithms?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
2
votes
1answer
69 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...
1
vote
3answers
42 views

Solve x in logarithm equation

I am trying to solve $x$ for $2log_{10} (x-4) - log_{10}4(x-1) = 0$ I have the key with the answer 10 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach ...
0
votes
3answers
44 views

Trouble with Logarithmic Differentiation

Hey guys I'm trying to find the derivative of this equation using logarithmic differentiation but I'm having some trouble. Wolfram Alpha is giving me different answers and I'm having difficulty ...
3
votes
2answers
145 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
1
vote
1answer
72 views

Why does $\log_{4}32 \neq \log _{4}(4 \cdot 8)$

$$\log_{4}32=2.5$$ If $$\log_{a}(b\cdot c) = \log _{a}b + \log_{a}c \,\,\,; (a>0, b>0,c>0, a\neq 1)$$ Then why does $\log_{4}32$ can't be $\log _{4}(4 \cdot 8)= \log_{4}4+\log_{4}8 = ...
1
vote
3answers
31 views

Solving for the value inside a base 10 logarithm

I have an equation of $\log(d)=(-x-A)/(10n)$ that I need to solve for $d$. How do I "reverse" the logarithm to obtain $d$? I apologize if this is super easy, I just can't even figure out how to Google ...
-1
votes
2answers
126 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
2
votes
2answers
52 views

solving equations with powers

Im trying to solve the equation $$3\cdot2^{-2/x} + 2\cdot9 ^{-1/x} = 5\cdot6^{-1/x }$$ So far I tried applying logaritmas but it didnt prove helpful...are there any other ways?
0
votes
3answers
62 views

Solve logarithmic equation: $2\log_7 (x+2) - \log_7 (3x+10) = 0$ [closed]

Please, can someone check if this is the right answer $$x= -2 \pm \sqrt{3x + 10}$$ Thank you.
2
votes
2answers
90 views

How do I simplify $\log (1/\sqrt{1000})$?

How do I simplify $\log \left(\displaystyle\frac{1}{\sqrt{1000}}\right)$? What I have done so far: 1) Used the difference property of logarithms $$\log ...
7
votes
3answers
202 views

Is $ln(x)$ ever greater than $x$

Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement? Just wondering for some convergence related thing
0
votes
1answer
31 views

Finding the exponent of $2$ such that $x \cdot 2^a$ is as close to $1$ as possible

How do I find an exponent of $2$ that when multiplied with another number would bring the result closest to the positive side $1$? Like this: $y = x \cdot 2^a$, where $y\ge 1$ has to be as small as ...
0
votes
2answers
37 views

Logorithms on a first level learning

Solve log$_{5x-1}$ $4$ $=$ $1/3$ $(5x-1)^{1/3}$=4 $((5x-1)^{1/3})^3$ = $4^3$ $5x-1=64$ $5x=65$ $13$ I am not sure where to go with this. I learned some things about logs before my class ended ...
1
vote
1answer
32 views

Help with Evaluating a Logarithm

A precalculus text asks us to evaluate $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{256}}{\sqrt[6]{32}}$ I do the following: $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{(2^2)^3\cdot 2^2}}{\sqrt[6]{2^3\cdot 2^2}}$ ...
1
vote
1answer
52 views

Solving $F(x) = 0.3$ where $F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}$

Consider the function $$F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}.$$ We want to solve $F(x) = 0.3$. Since $ F(x) = 0.3$ then we can say, $2.5 \ln (\frac{200}{x}) = 0.7$. $\ln(\frac{200}{x}) = 0 .28$. ...
2
votes
1answer
76 views

-ln(0.1) equalling to ln(10)?

I am having quite a headache wrapping my head around this solution. I do not understand the first line where they get lambda = ln(10) from statement to the left. Somebody please explain this to me. ...
0
votes
4answers
47 views

How do you solve this using only given values, logarithm rules and no calculator?

Given that $\log12=1.0792$ and $\log4=0.6021$, solve $\log8$ without a calculator. I am familiar with the following three rules: Product rule: $\log(a\cdot b)=\log a+\log b$ Quotient rule: ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
0
votes
1answer
70 views

How to simplify $3^{(2\log_335)}$

$3^{2\log_35}$ How do I simplify this? This is what I have done so far: $2\log_35=\log_35^2=\log_3(25)$ $3^{\log_3(25)}$ What do I do from here? And the answer is one of these mixed solutions: ...
0
votes
2answers
35 views

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$?

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$? There were some other questions like this and I applied this formula to them $\log_a(xy) = \log_a(x)+\log_a(y)$. However, in ...
3
votes
4answers
75 views

Solve for $x$ in $2\log(x+11)=(\frac{1}{2})^x$

Solve for $x$. $$2\log(x+11)=(1/2)^x$$ My attempt: $$\log(x+11)=\dfrac{1}{(2^x)(2)}$$ $$10^{1/(2^x)(2)}= x+11$$ $$x=10^{1/(2^x)(2)}-11$$ I'm not sure what to do next, because i have one $x$ in ...
1
vote
3answers
56 views

Why $\ln 2=\ln 1.075^t\implies \ln 2=t\ln 1.075$

Why $$\ln 2=\ln 1.075^t\implies \ln 2=t\ln 1.075$$
2
votes
1answer
29 views

Exponential continuous growth $\ln a$ vs. $r$? Huh?

So, given a simple population continuous growth problem, it seems that the entirety of the internet uses $P=P_0e^{rt}$ where $P$ is the population over time, $P_0$ is the initial population, $r$ is ...
2
votes
2answers
54 views

Expressing $\ln \sqrt[3]{54}$ in terms of $\ln 2$ and/or $\ln 3$

Express $\ln \sqrt[3]{54}$ in terms of $\ln 2$ and/or $\ln 3$ I know that $\sqrt[3]{54}=54^{1/3}$ but otherwise I don't know how to address these types of problems. How do I solve this, and is ...