If (log2 3 + log2 (6x - 4) = log2 (5 + 2x) + 3) what is the value of x

General
1

If (log2 3 + log2 (6x - 4) = log2 (5 + 2x) + 3) what is the value of x

2

If \log_2 3 + \log_2 (6x - 4) = \log_2 (5 + 2x) + 3, what is the value of x?

Solving Logarithmic Equations

To find the value of x, let’s solve the given logarithmic equation step by step.

Step 1: Combine Logarithms on the Left Side

Using the logarithmic property that states \log_b (A) + \log_b (B) = \log_b (A \cdot B), we can combine the terms on the left side of the equation:

\log_2 3 + \log_2 (6x - 4) = \log_2 [3(6x - 4)]

Thus the equation becomes:

\log_2 [3(6x - 4)] = \log_2 (5 + 2x) + 3

Step 2: Eliminate the Logarithms

We want to isolate the terms involving x. First, let’s handle the constant term +3 on the right side:

We can write +3 as \log_2 8 because 2^3 = 8, leading to:

\log_2 [3(6x - 4)] = \log_2 (5 + 2x) + \log_2 8

Again, use the logarithmic property to combine these logarithms on the right side:

\log_2 (5 + 2x) + \log_2 8 = \log_2 [8(5 + 2x)]

So the equation becomes:

\log_2 [3(6x - 4)] = \log_2 [8(5 + 2x)]

Since the logarithms are equal, we can set the arguments equal to each other because the logarithmic function is one-to-one:

3(6x - 4) = 8(5 + 2x)

Step 3: Simplify the Expression

Expand both sides of the equation:

  • Left side: 3(6x - 4) = 18x - 12
  • Right side: 8(5 + 2x) = 40 + 16x

Equate and simplify:

18x - 12 = 40 + 16x

Step 4: Solve for x

Move all x terms to one side and constant terms to the other:

18x - 16x = 40 + 12
2x = 52

Now, divide by 2:

x = \frac{52}{2}
x = 26

Verification

To verify the solution, substitute x = 26 back into the original equation:

Original expression:

\log_2 3 + \log_2 (6(26) - 4) = \log_2 (5 + 2(26)) + 3

Substitute x = 26:

  • \log_2 3 + \log_2 (156 - 4) = \log_2 (5 + 52) + 3
  • \log_2 3 + \log_2 152 = \log_2 57 + 3

Convert +3 into logarithmic form:

  • \log_2 8 = 3, so \log_2 3 + \log_2 152 = \log_2 57 + \log_2 8

Combine:

  • \log_2 (3 \times 152) = \log_2 (57 \times 8)

Calculate:

  • 3 \times 152 = 456 and 57 \times 8 = 456

Since both sides equal \log_2 456, the solution is correct.

Therefore, the value of x is indeed 26.

@anonymous7