# Thread: How to Solve for x in this Log Equation?

1. ## How to Solve for x in this Log Equation?

Hello,
I have no clue what property should be used for solving this problem or even how to solve it at all.
Here is the equation:
log5(log3x) = 0
How do I solve this?

2. ## Re: How to Solve for x in this Log Equation?

$\log_5(\log_3(x))=0$

$\log_3(x) = 5^0 = 1$

$x = 3^1 = 3$

$x=3$

3. ## Re: How to Solve for x in this Log Equation?

Originally Posted by Locopoco
Hello,
I have no clue what property should be used for solving this problem or even how to solve it at all.
Here is the equation:
log5(log3x) = 0
How do I solve this?
Suppose that each of $a~\&~b$ is a positive number.
$\log_a(B)=0\iff B=1$ AND $\log_b(C)=1\iff C=b$

Lets look at the posted question.
$\large\log_5(\log_3(x))=0\iff\log_3(x)=1\iff x=3$

4. ## Re: How to Solve for x in this Log Equation?

$\log_5(\log_3(x))=0$

Use identity,

$\log_ab=\dfrac{\log b}{\log a}$

$\Rightarrow\dfrac{\log\bigg(\dfrac{\log x}{\log(3)}\bigg)}{\log 5}=0$

$\Rightarrow\log\bigg(\dfrac{\log x}{\log(3)}\bigg)=0$

Taking exponent both sides

$\Rightarrow \dfrac{\log x}{\log 3}=1$

$\Rightarrow\log x=\log 3$

$\Rightarrow x=3$

5. ## Re: How to Solve for x in this Log Equation?

Here is a slightly different way that involves nothing more than $log_a(b) = c \iff b = a^c$ and substitution.

$Find\ x\ given\ log_5(log_3(x)) = 0.$

$Let\ u = log_3(x) \implies$

$log_5(u) = 0 \implies$

$u = 5^0 = 1 \implies$

$1 = u = log_3(x) \implies$

$x = 3^1 = 3.$

6. ## Re: How to Solve for x in this Log Equation?

You shouldn't worry about "properties" until you have a good grasp of "definitions"! And this problem is really about the definition of "log".