Question

How do you use the Change of Base Formula and a calculator to evaluate the logarithm \[{\log _5}7\]?

Answer 1

Hint: A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e. The Change of Base Formula is given as: \[{\log _b}a = \dfrac{{{{\log }_c}a}}{{{{\log }_c}b}}\], in which the value of a is 7 and b is 5, hence simplify by applying the log values using calculator to the given values.

Formula used: \[{\log _b}a = \dfrac{{{{\log }_c}a}}{{{{\log }_c}b}}\]

Complete step by step solution:
 Let us write the given data,
The change of base formula states that:
\[{\log _b}a = \dfrac{{{{\log }_c}a}}{{{{\log }_c}b}}\]
So, therefore according to the given logarithm \[{\log _5}7\] we get:
\[{\log _5}7 = \dfrac{{\log 7}}{{\log 5}} = \dfrac{{\ln 7}}{{\ln 5}}\]
\[{\log _5}7 = \dfrac{{1.94591}}{{1.609438}}\]
\[{\log _5}7 = 1.2090617\]
\[{\log _5}7\mathop ~ 1.21\]

Therefore the answer is \[{\log _5}7 ~ 1.21\]

Additional information: The logarithmic function is also defined by,
if \[{\log _a}b = x\], then \[{a^x} = b\].
Where x is defined as the logarithm of a number ‘b’ and ‘a’ is the base of the log function that could have any base value, but usually, we consider it as ‘e’ or ‘10’ in terms of the logarithm. The value of the variable ‘a’ can be any positive number but not equal to 1 or negative number.

Note: We must note that \[\log 1 = 0\] and \[\log 10 = 1\]. It can be evaluated using the logarithm function, which is one of the important mathematical functions. Log functions are commonly used to solve many lengthy problems and reduce the complexity of the problems by reducing the operations from multiplication to addition and division to subtraction. The log function with base 10 is called the common logarithmic functions and the log with base e is called the natural logarithmic function.