Question

What is the value of x in 2(5^x)=14?

A. log5/log7
B. log7-log5
C. log7/log5
D. log2

Answer 1

`\textsf{C)} \quad (\log7)/(\log5)`

Explanation:

Given equation:

`2\left(5^x\right)=14`

Isolate the exponential term by dividing both sides of the equation by 2:

`(2\left(5^x\right))/(2)=(14)/(2)`

`5^x=7`

`\textsf{Apply the log law:} \quad a^c=b \iff \log_ab=c`

`\log_57=x`

`x=\log_57`

`\textsf{Apply the change of base rule}, \; \log_ab=(\log_nb)/(\log_na),\;\textsf{where the new base is 10:}`

`x=\log_57=(\log_(10)7)/(\log_(10)5)`

As log without a specified base is usually assumed to be the logarithm with base 10, we can write the value of x as:

`x=(\log7)/(\log5)`