Question

for any base a >o,1prove that loga(m)m=n loga m​

Answer 1

Answer:

Logarithms, like exponents, have many helpful properties that can be used to simplify logarithmic expressions and solve logarithmic equations. This article explores three of those properties.

Let's take a look at each property individually.

The product rule: \log_b(MN)=\log_b(M)+\log_b(N)log

b

(MN)=log

b

(M)+log

b

(N)log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis

This property says that the logarithm of a product is the sum of the logs of its factors. Show me a numerical example of this property please.

We can use the product rule to rewrite logarithmic expressions.

Example: Expanding logarithms using the product rule

For our purposes, expanding a logarithm means writing it as the sum of two logarithms or more.

Let's expand \log_6(5y)log

6

(5y)log, start base, 6, end base, left parenthesis, 5, y, right parenthesis.

Notice that the two factors of the argument of the logarithm are \blueD 55start color #11accd, 5, end color #11accd and \greenD yystart color #1fab54, y, end color #1fab54. We can directly apply the product rule to expand the log.

\begin{aligned} \log_6(\blueD5\greenD y)&=\log_6(\blueD5\cdot \greenD y) \\\\ &=\log_6(\blueD5)+\log_6(\greenD y)&&{\gray{\text{Product rule}}} \end{aligned}

log

6

(5y)

=log

6

(5⋅y)

=log

6

(5)+log

6

(y)

Product rule