Question

How do you write `log_14 196=2` in exponential form?

Answer 1

`14^2=196`

Explanation:

Suppose you had `log_ax=y`

They would call this log to base `a`

Then it means: `a^y=x` .............................Equation(1)

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Note that `log_a(a)=1`

So:
`log_2(2)=1" "->" "2^1=2`
`log_10(10)=1" "->" "10^1=10`
`log_e(e)=1" "->" "e^1=e`

`Log_e` is a special one case.

You normally see this written as `ln`

So you could have `ln(x)` sometimes you see it as `exp(x)`

The last one is quite often used in computer software and in higher maths.

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Note that `log_10(1)=0` as is any `log_x(1)=0`

Look at Equation(1) and you will observe that

as in `log_ax=y" "->" "a^y=x`

`" "log_10(1)=0" "->" "10^0=1`

Any value (apart from 0) raised to the power of 0 has the value 1

Consider: `x^z/x^z =1` this is the same as `x^(z-z) =x^0=1`

Answer 2

`log_14 196 = 2" " rArr " " 14^2 = 196`

Explanation:

Changing from log from to index form can be done by purely following the definition.

`log_a b = c " " rArr " a^c = b`

Remember:
"The base stays the base and the other two change around"

`log_14 196 = 2" " rArr " " 14^2 = 196`