Question 71846
The log of base 9 goes on both sides. The log undoes an exponent, so we use the log on the entire right side, but we must do it to both sides.
{{{5=9^(x-1)}}}
{{{log_[9](5)=log_[9](9^(x-1))}}}Take the log of both sides.The log simply cancels out the base 9
{{{log_[9](5)= x-1}}}Add 1 to both sides
{{{log_[9](5)+1=x}}}Since log_[9](5) is a number we can find x
Use the change of base formula
{{{log_[b](x)=log(x)/log(b)}}}Where b is the base and x is the argument. Remember the base of the log is assumed to be 10 if not specified.
{{{x=log(5)/log(9)+1}}}So this can be evaluated on a calculator
{{{x=0.69897/0.95424 + 1}}}
{{{x=0.73248 + 1}}}
{{{x=1.73248}}}
Check:
{{{5=9^(1.73248-1)}}}
{{{5=9^(0.73248)}}}
{{{5=4.99993}}} Which is close enough to 5 (there are some roundoff errors).