Question

Solve for \[x\] if \[{4^x} = 13\].

Answer 1

Hint: Since there is exponent on the left hand side, try to take log on both sides to solve the problem. Then use properties of log like \[\log {a^b} = b\log a\], \[\log ab = \log a + \log b\], \[\log \dfrac{a}{b} = \log a - \log b\], simplify both sides.

Complete step by step solution:
We have to solve for \[x\] in \[{4^x} = 13\]
Since there is an exponent on one side, we take logs on both sides, to solve the problem.
Taking \[{\log _{10}}\]on both sides we get,
\[{\log _{10}}{4^x} = {\log _{10}}13\]
Now using the property \[\log {a^b} = b\log a\], on L.H.S.,
\[ \Rightarrow x{\log _{10}}4 = {\log _{10}}13\]
\[ \Rightarrow x{\log _{10}}{2^2} = {\log _{10}}(10)\]
\[ \Rightarrow 2x{\log _{10}}2 = {\log _{10}}13\]
\[ \Rightarrow 2x = \dfrac{{{{\log }_{10}}13}}{{{{\log }_{10}}2}}\]
Using the property \[\dfrac{{{{\log }_c}b}}{{{{\log }_c}a}} = {\log _a}b\]on R.H.S., we get
\[ \Rightarrow 2x = {\log _2}13\]
\[ \Rightarrow x = \dfrac{{{{\log }_2}13}}{2}\]
\[ \Rightarrow x = \dfrac{{3.7004}}{2} = 1.8502\]

Note: For solving logarithm problems the standard log values of \[{\log _{10}}2,{\log _{10}}3,{\log _{10}}10\], must be memorized. Also one must always try to apply the logarithm properties for solving the problems. Some of the frequently used properties are:
\[{\log _{10}}\left( {a \cdot b} \right) = {\log _{10}}a + {\log _{10}}b\]
\[{\log _{10}}\left( {\dfrac{a}{b}} \right) = {\log _{10}}a - {\log _{10}}b\]
\[{\log _{10}}{a^b} = b{\log _{10}}a\]
\[\dfrac{{{{\log }_c}b}}{{{{\log }_c}a}} = {\log _a}b\]