Question

Log0,64(3x-50)=log1,25 0,5

Answer 1

ОДЗ : 3x - 50 > 0     ⇒   3x > 50    ⇒    x > 16 2/3

$log_{0,64}(3x-50)=log_{1,25}0,5\\\\log_{(\frac{4}{5})^{2}}(3x-50)=log_{(\frac{4}{5})^{-1}}0,5\\\\log_{\frac{4}{5} }(3x-50)^{\frac{1}{2}}=log_{\frac{4}{5} }(\frac{1}{2})^{-1}\\\\\sqrt{3x-50}=2\\\\(\sqrt{3x-50})^{2}=2^{2}\\\\3x-50=4\\\\3x=54\\\\x=18$

Answer 2

To solve the equation, we need to use logarithmic properties to eliminate the logarithms and then solve for the variable 'x'.

The given equation is:

log₀,₆₄(3x - 50) = log₁,₂₅(0.5)

Step 1: Use the change of base formula to convert both logarithms to a common base, such as base 10.

log₀,₆₄(3x - 50) = log₁,₂₅(0.5)

log₁₀(3x - 50) / log₁₀(64) = log₁₀(0.5) / log₁₀(1.25)

Step 2: Calculate the logarithms using a calculator.

log₁₀(3x - 50) / log₁₀(64) = -0.3010 / 0.0969

Step 3: Simplify the equation.

log₁₀(3x - 50) / 1.806 = -3.105

Step 4: Now, isolate the logarithm expression on one side.

log₁₀(3x - 50) = -3.105 * 1.806

log₁₀(3x - 50) = -5.609

Step 5: Remove the logarithm by exponentiating both sides with base 10.

10^(-5.609) = 3x - 50

Step 6: Solve for 'x'.

0.000002139 ≈ 3x - 50

3x ≈ 50.000002139

x ≈ 50.000002139 / 3

x ≈ 16.666667

So, the approximate value of 'x' is 16.67.