Problems on Logarithms into Exponentials

Algebra > Logarithms > Solved Examples

2.Let us now solve a few problems applying the logarithm formulas and concepts learned so far.

1. Change the following exponential forms into logarithmic forms.
1. 23 = 8         2. 3√(64) = 4          3. (√2)8 = 16

Solution:

1. we know that if

ax = n, then log a n = x

therefore, if

23 = 8, then log 2 8 = 3

2. 3√(64)

= (64)1/3 = 4

log 64 4 = 1/3

3. (√2)8 = 16

log √2 16 = 8

2. Change the following logarithmic forms into exponential forms:

1. log 4 16 = 2         2. log 6 36 = 2      3. log a 3√(x) = 1/y

Recall that if

log a x = n, then an = x

Solution1:

log 416 = 2, 42 = 16

Solution 2:

log 6 36 = 2, 62 = 36

Solution 3:

log a 3√(x) = 1/y ,

a1/y = 3√(x)

a1/y = x1/3

2. Find log 2 (log 3 81) = log 3 X

Solution:

Let log 3 81 = n,

then 3n = 81,

3 n = 34 ,

n = 4

so, in the given question,

log 2 (log 381) reduces to log 2 4.

Now the question is

log 2 4 = log 3 X…………………..(2)

again, let

log 2 4 = n,

then 2n = 4 = 22, so n = 2

writing n = 2 in (2), we get

2 = log 3 X

So, 32 = X

X = 9

Now, we will learn simple laws of logarithms to solve above type of problems easily and in far fewer steps in the following pages.