Question

How do you express a sum of logarithms for \[{\log _5}\left( {125 \cdot 25} \right)\]?

Answer 1

Hint: Here the given function is a logarithm function it can be defined as logarithmic functions are the inverses of exponential functions. The given log function has base value 5 by using the Basic Properties of logarithmic function. We can express the given logarithm function into a sum of logarithmic functions.

Complete step-by-step solution:
The function from positive real numbers to real numbers to real numbers is defined as \[{\log _b}:{R^ + } \to R \Rightarrow {\log _b}\left( x \right) = y\], if \[{b^y} = x\], is called logarithmic function or the logarithm function is the inverse form of exponential function. In logarithmic function basically we have two kinds common logarithm function and natural logarithm function. Here the question has log term and its base is 5. So it does not comes under any of two kinds. But the properties are applicable to the log function

There are some basic logarithms properties
1. product rule :- \[{\log _b}\left( {mn} \right) = {\log _b}m + {\log _b}n\]
2. Quotient rule :- \[{\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}m - {\log _b}n\]
3. Power rule :- \[{\log _b}\left( {{m^n}} \right) = n.{\log _b}m\]
Now, Consider the given logarithm function, it has base 5.
 \[ \Rightarrow \,\,\,{\log _5}\left( {125 \cdot 25} \right)\] ------(1)
By using the product rule of logarithm properties
Where base b=5, m=125 and n=25
Then, equation (1) can be rewritten as
\[ \Rightarrow \,\,\,{\log _5}\left( {125 \cdot 25} \right) = {\log _5}125 + {\log _5}25\] ------ (2)

Hence, the sum of logarithmic functions is \[{\log _5}\left( {125 \cdot 25} \right) = {\log _5}125 + {\log _5}25\].

Note: The question contains the log terms we must know the logarithmic properties which are the standard properties. The properties are on the addition, subtraction, multiplication, division and exponent. The base value may differ but the properties will be the same, it won’t alter.