(iv) Log_4)(3^(x)-1)log_(1/4)(3^(x)-1)/(16)

Base Change Formula: log_a(b) = log_c(b)/log_c(a)
Inverse Property: log_a(a^b) = b
Product Rule: log_a(b) + log_a(c) = log_a(bc)

3 min read Jun 16, 2024

Solving the Logarithmic Equation: log_4(3^(x)-1)log_(1/4)(3^(x)-1)/(16)

This article will explore the solution to the logarithmic equation:

log_4(3^(x)-1)log_(1/4)(3^(x)-1)/(16)

To solve this equation, we will utilize several key properties of logarithms:

Simplify the Equation:

Let's start by simplifying the expression using the base change formula. We can rewrite log_(1/4)(3^(x)-1) in terms of base 4:

log_(1/4)(3^(x)-1) = log_4(3^(x)-1)/log_4(1/4) = -log_4(3^(x)-1)

Substituting this back into the original equation:

log_4(3^(x)-1) * (-log_4(3^(x)-1))/16
Simplify further:
- (log_4(3^(x)-1))^2 / 16
Solve for the expression:

Let y = log_4(3^(x)-1). The equation becomes:

-y^2/16 = 0

Multiplying both sides by -16:

y^2 = 0

Taking the square root of both sides:

y = 0
Substitute back and solve for x:

Now, substitute y back:

log_4(3^(x)-1) = 0

Using the inverse property of logarithms:

3^(x)-1 = 4^0 = 1

Adding 1 to both sides:

3^(x) = 2

Solving for x by taking the logarithm of both sides (using any base):

x * log(3) = log(2)

x = log(2)/log(3)

Therefore, the solution to the logarithmic equation log_4(3^(x)-1)log_(1/4)(3^(x)-1)/(16) is:

x = log(2)/log(3)

This solution is a single value.
We can approximate the value of x using a calculator.
Always check for any restrictions on the domain of the original equation to ensure the solution is valid.