What is the approximate value of x in the equation log34(25) + 3x = 1?

To find the approximate value of x in the equation log₃₄(25) + 3x = 1, we can start by isolating the variable x.

1. First, we can rewrite the equation:

3x = 1 – log₃₄(25)

2. Next, we need to calculate the value of log₃₄(25). This logarithm asks the question: to what power must 34 be raised to produce 25? To calculate this, we can use the change of base formula for logarithms:

log_b(a) = log_k(a) / log_k(b)

Using the natural logarithm (base e):

log₃₄(25) = ln(25) / ln(34)

Calculating the natural logarithms:

• ln(25) ≈ 3.2189
• ln(34) ≈ 3.526

Plugging these values into our formula gives us:

log₃₄(25) ≈ 3.2189 / 3.526 ≈ 0.9141

3. Now, substitute back into the equation:

3x = 1 – 0.9141

4. This simplifies to:

3x ≈ 0.0859

5. Finally, divide both sides by 3 to isolate x:

x ≈ 0.0859 / 3 ≈ 0.0286

So, the approximate value of x in the equation log₃₄(25) + 3x = 1 is x ≈ 0.0286.