Question

I am to solve the exponential equation of 5^(x+3)=242 in natural logarithms.
I have so far for the solution set ln 5((x+3))/ln 242
I need help better understanding how to do this and on how to use a calculator to give the decimal approximation for the set.

Answer 1

start with
\[ 5^{(x+3)}=242 \]
"take the ln" of both sides
\[ \ln\left( 5^{(x+3)}\right) = \ln (242) \]
you can "bring out" the exponent on the left side to get
\[ (x+3) \ln(5) = \ln(242) \]
I would divide both sides by ln(5) as the next step.

Answer 2

\(\bf recall\implies log(x^{\color{red}{ a}})\to {\color{red}{ a}}log(x)\)

Answer 3

\(\bf 5^{(x+3)}=242\implies ln\left[5^{{\color{red}{ (x+3)}}}\right]=ln(242)\implies {\color{red}{ (x+3)}}ln(5)=ln(242)\)

as phi pointed out

Answer 4

We are asked to give a solution set in terms of natural logarithms- is it supposed to look like this in the end? \[\frac{ \ln 242 }{ \ln (5((x+3)) }\]