Question

Write $2\log 3 + 3\log 5 + 5\log 2$ as a single logarithm.

Answer 1

Hint : Logarithms can be defined as the ways to figure out which exponents and when we need to multiply to get the specific number. Here we will use product rules and the power to simplify the given expression.

Complete step-by-step answer :
The logarithm is defined as the power for which the number must be raised in order to get some other terms. Always remember the standard and the basic properties of the logarithm such as Product rule, quotient rule and the power rule. The basic and appropriate logarithm properties are most important since the solution totally depends on it, so remember and understand its application properly.
Take the given expression: $2\log 3 + 3\log 5 + 5\log 2$
Here apply, Power rule: $\log {x^n} = n\log x$ in the above expression –
$= \log {3^2} + \log {5^3} + \log {2^5}$
Simplify the above expression finding the powers of the terms –
$= \log 9 + \log 125 + \log 32$
Now, Apply Product rule: $\log xy = \log x + \log y$ for all the three terms
$= \log (9 \times 125 \times 32)$
Simplify finding the product of the terms –
$= \log 36000$
This is the required solution.
So, the correct answer is “$\log 36000$ ”.

Note : Also refer to the below properties and rules of the logarithm.
Product rule: ${\log _a}xy = {\log _a}x + {\log _a}y$
Quotient rule: ${\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y$
Power rule: ${\log _a}{x^n} = n{\log _a}x$
Base rule: ${\log _a}a = 1$
Change of base rule: ${\log _a}M = \dfrac{{\log M}}{{\log N}}$