Question

How do you solve \[{{\log }_{b}}64=3\]?

Answer 1

Hint: In this problem we have to evaluate and find the logarithmic value of \[{{\log }_{b}}64=3\]. We can first write the given logarithmic expression in exponential form using the definition of logarithm. We know that the logarithmic formula to solve this problem, comparing the formula and the given expression, we can get the value of b, x which can be substituted in the logarithmic formula. We will get the same base terms which should be cancelled to get the value for the given logarithmic expression.

Complete step by step answer:
We know that the given logarithmic expression is \[{{\log }_{b}}64=3\].
Now we can rewrite the above expression in exponential using the definition of logarithm,
\[{{\log }_{b}}64=3\] …. (1)
We also know that if x and b are positive real numbers and does not equal to 1, then
\[{{\log }_{b}}x=y\] is equivalent to \[{{b}^{y}}=x\]…. (2)
Now we can compare the logarithmic expression (1) to the above expression, we get
y = 3, x = 64.
Now we can substitute the above values in the expression (2), we get
\[\begin{align}
  & \Rightarrow {{b}^{3}}=64 \\
 & \Rightarrow {{b}^{3}}={{4}^{3}}\text{ }\because {{4}^{3}}=64 \\
\end{align}\]
Since we have same base terms in the above step, we can cancel them to get
\[\Rightarrow b=4\]

Therefore, by solving \[{{\log }_{b}}64=3\], the value of b is 4.

Note: Students make mistakes while writing the correct logarithmic formula to evaluate these types of problems. To solve these types of problems, we should know basic logarithmic formulas and understand the concept and properties of logarithm.