Question

Solve 3^(x+1) = 15 for x using the change of base formula

Answer 1

Answer:

$x=\frac{log(15)}{log(3)}-1$

Step-by-step explanation:

$3^{x+1} = 15$

LEts convert exponential form to log form

$b^x=a$ can be written as $log_b(a)=x$

WE apply the same rule to convert the given exponential form to log form

$3^{x+1} = 15$

$log_3{15} = x+1$

HEre the base of log is 3. Lets apply change of base formula

$log_b(a)=\frac{log(a)}{log(b)}$

$log_3{15} = x+1$

$\frac{log(15)}{log(3)} = x+1$

Now subtract 1 from both sides

$x=\frac{log(15)}{log(3)}-1$

Answer 2

$\bf \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \\\\\\ \textit{Logarithm Change of Base Rule} \\\\ \log_a b\implies \cfrac{\log_c b}{\log_c a}\qquad \qquad c= \begin{array}{llll} \textit{common base for }\\ \textit{numerator and}\\ denominator \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf 3^{x+1}=15\implies \log_{10}(3^{x+1})=\log_{10}(15)\implies (x+1)\log_{10}(3)=\log_{10}(15) \\\\\\ x+1=\cfrac{\log_{10}(15)}{\log_{10}(3)}\implies \stackrel{\textit{change of base rule}}{x=\cfrac{\log_{e}(15)}{\log_{e}(3)}-1}\implies x\approx 1.47$