# Rewrite $3^{4x-5}=38$ as common log and then use a calculator to solve

I am given the expression $$3^{4x-5}=38$$ and asked to rewrite in common log to isolate the variable $$x$$, then to solve using a calculator.

I am struggling to to get the $$x$$ on it's own. My attempt:

$$3^{4x-5}=38$$

Rewrite lhs as a log

$$log_3(4x-5)=38$$

Rewrite lhs into common log base 10:

$$\frac{log(4x-5)}{log(3)}=38$$

This is as far as I can go. How can I isolate x here?

I rewrote to this, does it look right?

$$3^{4x-5}=38$$ $$log_3(38)=4x-4$$

My textbook says I am specifically to rewrite using the common log: $$\frac{log(38)}{log(3)}+5=4x$$ $$x=\frac{(\frac{log(38)}{log(3)}+5))}{4}$$

• $a^b=c$ gives $b\ln(a)=\ln(c)$ so there is a mistake both on LHS and RHS. Equivalently you could have written $\log_3(38)=4x-5$. – zwim Jan 23 at 14:42
• After your edit, this is correct. – zwim Jan 23 at 15:59

## 1 Answer

Hint: $$\log a^b = b \log a$$ and not $$\log_ba$$