# Evaluating the logarithms of surds

My strategy for evaluating the logarithms of surds that are in the form of: log$$_b$$ ($$^x\surd$$rad) = y involves transforming the surd to exponent form, where the radicand becomes the base and the index is the reciprocal of the surds’ and putting the whole thing into what I think of as index notation (b$$^x$$= y)

For example, evaluating log$$_3$$ ($$^4\surd$$9) = x converts to log$$_3$$ (9$$^{1/4}$$) = x

In the next stage of the process, I rearrange the initial statement to the form b$$^x$$ = y and perform further operations on the Left Hand Side (LHS) to deduce x.

For example:
step 0: 3$$^x$$ = 9$$^{1/4}$$
step 1: (3$$^2$$)$$^x$$ = 9$$^{1/4}$$
step 2: (3$$^2$$)$$^{1/4}$$ = 9$$^{1/4}$$
step 3: (3$$^{2/4}$$) = 9$$^{1/4}$$
step 4: (3$$^{1/2}$$) = 9$$^{1/4}$$
… final answer: log$$_3$$ ($$^4\surd$$9) = 1/2

What’s got me confused is that I arrived at the process in the second half by a flash of insight/intuition and am struggling to determine how the operations I perform on the LHS are legal, since in step 1 I appear to change the value of one side without changing the value of the other side.

I can only conclude that my understanding of equation transposition must be flawed. Request for comment.

• Cye Waldman's comment is definitely correct. $3^x = 9^{1/4}$. Taking the natural logarithm of each side gives $\ln 3^x = \ln 9^{1/4}$. Since $\ln b^t = t\ln b$ whenever $b > 0$ and $b \neq 1$, $x\ln 3 = \frac{1}{4}\ln 9$. Since $9 = 3^2$, $x\ln 3 = \frac{1}{4} \ln 3^2$. Using the rule $\ln b^t = t\ln b$ again, gives $x\ln 3 = \frac{1}{2}\ln 3$. Since $\ln 3 \neq 0$, we may divide both sides of the equation by $\ln 3$ to obtain $x = 1/2$. Commented Apr 18, 2022 at 9:31

I think the confusion is in how you write the steps. In particular, I am confused by how you went from step 0 to 1 as well. Let me rephrase what I think you are trying to say, and see if it makes more sense.

Step 0: We want to solve $$3^x = 9^{\frac{1}{4}}$$.

Step 1: We know that $$9 = 3^2$$, so $$\color{red}{\text{RHS}} = (3^2)^{\frac{1}{4}}$$, meaning $$3^x = (3^2)^{\frac{1}{4}}$$.

Step 2: Now, we follow your step $$3$$ and $$4$$ to get $$\color{red}{\text{RHS}} = 3^{\frac{1}{2}}$$, meaning $$3^x = 3^{\frac{1}{2}}$$.

Finally, this means $$x = \frac{1}{2}$$.

Does this help?

• Yes… this is it. Thank you Gareth. That’s consistent with my learning to date. The process of equating indexes requires that the bases be made similar first, as you do. Commented Apr 17, 2022 at 23:29