Finding values of x for logarithm

Question

The question is to find the numbers of x which satisfy the equation.

$$ \log_x10=\log_4100. $$

I have \begin{align*} \frac{\ln10}{\ln x} &= \frac{\ln 100}{\ln 4} \\ \frac{\ln10}{\ln x} &= \frac{2\ln 10}{2\ln 2} \end{align*}

What would I do after this step?

Answer 1

We want to find an $x$ such that $$\log_x10=\log_4100$$ is satisfied.

Continuing from your steps:

$\frac{\ln10}{\ln x}=\frac{\ln100}{\ln4}$

$\frac{\ln10}{\ln x}=\frac{2\ln10}{2\ln2}$

$\ln x=\frac{(2\ln2)(\ln10)}{2\ln10}$

$\ln x=\ln2$

$x=2$

Answer 2

All right. First multiply by $ln(x)$ and by $ln(2)$. You get

$ln(10)ln(2)=ln(10)ln(x)$

Now divide by $ln(10)$. This gives you

$ln(2)=ln(x)$.

Now you apply the exponential function on both sides to get rid of the logarithm:

$\underbrace{e^{ln(x)}}_{=x}=\underbrace{e^{ln(2)}}_{=2}$

So $x=2$