find x-intercept of a logarithmic function

Question

Find the x-intercept of:

$3-4\log _{10}\left(2x\right) $

My process:

$4\log _{10}\left(2x\right)=3 $

$\log _{10}\left(2x\right)\:=\:\frac{3}{4}$

How would I find the x-intercept from here?

$\log _{10}\left(2x\right)\:=\: \log _{10}\left(?\right)\:$

$2x=?$

$xintercept=\frac{?}{2}$

How do I find the x-intercept?

Answer 1

you can write $$\log_{10}(2x)=\log_{10}10^{3/4}$$

Answer 2

Let $$f(x)=3-4\log _{10}\left(2x\right) $$ and set$$f(x)=0$$ which gives

$$3=4\log _{10}(2x)$$ $$\frac34=\log _{10}(2x)$$ Now $\log_ab=c$ implies $a^c=b$ or like you have done $\log_{10}(2x)=\log_{10}10^{3/4}$ which gives $$10^{3/4}=2x$$ $$x=\frac{10^{3/4}}{2}$$