Question

How do you solve \[x = {\log _{12}}144\]?

Answer 1

Hint: When one term is raised to the power of another term, the function is called an exponential function, for example \[a = {x^y}\] . The inverse of the exponential functions are called logarithm functions, the inverse of the function given in the example is\[y = {\log _x}a\] that is a logarithm function. Here we need to solve for ‘x’.

Complete step-by-step solution:
Given \[x = {\log _{12}}144\],
We know that \[{\log _a}\left( b \right) = x\] means that ‘x’ is the exponent that you must give to a to obtain b.
That is \[{a^x} = b\]
Now for the given problem we will have,
\[{12^x} = 144\]
we know that for only \[x = 2\] we will have 144.

Hence the solution of \[x = {\log _{12}}144\] is \[x = 2\].

Note: Product rule of logarithm that is the logarithm of the product is the sum of the logarithms of the factors. That is \[\log (x.y) = \log (x) + \log (y)\]. Quotient rule of logarithm that is the logarithm of the ratio of two quantities is the logarithm of the numerator minus the logarithm of the denominator. that is \[\log \left( {\dfrac{x}{y}} \right) = \log x - \log y\]. Power rule of logarithm that is the logarithm of an exponential number is the exponent times the logarithm of the base. That is \[\log {x^a} = a\log x\]. These are the basic rules we use while solving a problem that involves logarithm function.