Question

Solve ${3^x} - 2 = 11$ ?

Answer 1

Hint: First simplify the equation ${3^x} - 2 = 11$ by transferring $2$ to the right-hand side of the equation.
Take the logarithm of each side of the equation.
The solution of the equation is in the logarithmic function.

Complete step by step answer:
Consider the given equation is ${3^x} - 2 = 11$.
Add $2$ to each side of the equation.
${3^x} - 2 + 2 = 11 + 2$
$ \Rightarrow {3^x} = 13$
Take the logarithm each side of the equation,
$\log \left( {{3^x}} \right) = \log 13$
Apply the property of the logarithmic function, that is, $\log {a^b} = b\log a$,
$ \Rightarrow x\log 3 = \log 13$
$ \Rightarrow x = \dfrac{{\log 13}}{{\log 3}}$

Note: A logarithm is the opposite of a power.
$\log (xy) = \log x + \log y$
$\log \left( {\dfrac{x}{y}} \right) = \log x - \log y$
$\log \left( {{x^y}} \right) = y\log x$
$\log e = 1$
\[\log (1) = 0\]
We can calculate that \[{10^3} = 1000\] , we know that ${\log _{10}}1000 = 3$ (“log base \[10\] of \[1000\] is $3$ ”). Using base \[10\] is common.
Use exponents with base e, it's even more natural to use e for the base of the logarithm. This natural logarithm is frequently denoted by\[\;\ln (x)\] , i.e.,
\[\ln (x) = {\log _e}x\]