3
$\begingroup$

I've been trying to solve a problem:

$3^{1-x}=2$

I converted this to a log as: $\log_{3}{2} = (1-x)$

But I couldn't see how to progress from there. Having put it into a solving team, it suggests that it can be translated to: $\left(1-x\right)\ln \left(3\right)=\ln \left(2\right)$

...but I can't see why this makes sense. Could anyone point me in the right direction?

$\endgroup$
  • 1
    $\begingroup$ "But I couldn't see how to progress from there". You have an equation with one unknown, namely $x$. Re-arrange to make $x$ the subject of the equation [ Hint: the brackets don't serve a purpose in the equation:$log_{3}{2} = (1-x)\ $] $\endgroup$ – Adam Rubinson Nov 29 '20 at 0:49
  • $\begingroup$ @AdamRubinson Thanks - I'll do that, appreciate the useful advice on how to reach the solution $\endgroup$ – user133935 Nov 29 '20 at 0:51
3
$\begingroup$

This is due to the change of base formula, $$\log_{b}x = \frac{\ln x}{\ln b}.$$

This is because if you have $\log_{b}x = a$, then you can rewrite it as $x = b^{a}$. Taking the natural logarithm of each side gives you $\ln x = \ln (b^a)$, which can be rewritten as $\ln x = a\ln b$, which upon dividing both sides by $\ln b$ gives you $a = \frac{\ln x}{\ln b}$. Hence, you have $$\log_{b}x = a = \frac{\ln x}{\ln b}.$$

$\endgroup$
  • 1
    $\begingroup$ Fantastic, thank you - I will look this up $\endgroup$ – user133935 Nov 29 '20 at 0:48
1
$\begingroup$

Use the property of logarithms that $\log_b x=\dfrac {\log_e x}{\log_e b}$.

$\endgroup$
  • 1
    $\begingroup$ Fantastic, thank you - I will look this up $\endgroup$ – user133935 Nov 29 '20 at 0:48
  • $\begingroup$ @user133935: This answer is wrong for $e=1$. $\endgroup$ – user21820 Nov 29 '20 at 9:58
  • $\begingroup$ The base of natural logarithms $e\ne1$ $\endgroup$ – J. W. Tanner Nov 29 '20 at 13:13
0
$\begingroup$

If you mean to find the value of $x$ you almos have it. Another option is to write $3^{(1-x)}=2$ as $\frac{3}{3^x}=2$ which is equivalent to $\frac{3}{2}=3^x$ and using $log_3$ the value of $x$ is $log_3\frac{3}{2}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.