Question

If log₁₀ (3 x - 5) - log₁₀ (4 x - 3) = log₁₀ 25, find the value of x.

Answer 1

Answer:

In convert Exponentials and Logarithms we will mainly discuss how to change the logarithm expression to Exponential expression and conversely from Exponential expression to logarithm expression.

To discus about convert Exponentials and Logarithms we need to first recall about logarithm and exponents.

The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number. Thus, if aˣ = N, x is called the logarithm of N to the base a.

For example:  

1. Since 3⁴ = 81, the logarithm of 81 to base 3 is 4.  

2. Since 10¹ = 10, 10² = 100, 10³ = 1000, ………….

The natural number 1, 2, 3, …… are respectively the logarithms of 10, 100, 1000, …… to base 10.  

The logarithm of N to base a is usually written as log₀ N, so that the same meaning is expressed by the two equations  

ax = N; x = loga N

Examples on convert Exponentials and Logarithms

1. Convert the following exponential form to logarithmic form:

(i) 104 = 10000

Solution:

104 = 10000

⇒ log10 10000 = 4

(ii) 3-5 = x

Solution:

3-5 = x

⇒ log3 x = -5

(iii) (0.3)3 = 0.027

Solution:

(0.3)3 = 0.027

⇒ log0.3 0.027 = 3

2. Convert the following logarithmic form to exponential form:

(i) log3 81 = 4

Solution:

log3 81 = 4

⇒ 34 = 81, which is the required exponential form.

(ii) log8 32 = 5/3

Solution:

log8 32 = 5/3

⇒ 85/3 = 32

(iii) log10 0.1 = -1

Solution:

log10 0.1 = -1

⇒ 10-1 = 0.1.

hope it helps :)