How do you solve \[{12^r} = 13\]?

Hint: Here we have to solve the above equation. The above equation is in the form of exponential form. The exponential number is defined as the number of times we multiply the number by itself. So, we can’t solve this directly so we apply log to it and solve further.

Complete step-by-step solution:
The logarithmic function and the exponential function are both inverse of each other. The exponential number can be written in the form of a logarithmic number and likewise we can write the logarithmic number in the form of an exponential number.

Now we have equation \[{12^r} = 13\]
Apply log on the both sides we have
\[
   \Rightarrow \log \left( {{{12}^r}} \right) = \log \left( {13} \right) \\
   \Rightarrow \log {12^r} = \log 13 \\
 \]
By the logarithmic property we have \[\log {a^m} = m\log a\], using this property we have
\[ \Rightarrow r\log 12 = \log 13\]
Now we write the equation for r so we have
\[ \Rightarrow r = \dfrac{{\log 13}}{{\log 12}}\]
The logarithm in the above inequality is a common logarithmic function. Since the base value for the log is not mentioned. It is obviously considered the base value as 10.
By using the Clark’s table finding each term value and then on simplification we get
\[ \Rightarrow r \approx 1.032\], nearly.
Suppose in the above inequality if the common logarithmic function is replaced by the natural logarithmic function then the value will change.

Note: The exponential number is inverse of logarithmic. But here we have not used this. We have applied the log on both terms. The logarithmic functions have several properties on addition, subtraction, multiplication, division and exponent. So we have to use logarithmic properties. We have exact values for the numerals by the Clark’s table, with the help of it we can find the exact value.