Math Problem Statement

Solve: log 8x - log 2 = log 32

Solution

Let's solve the equation:

log8xlog2=log32\log 8x - \log 2 = \log 32

Step-by-step:

  1. Apply the logarithmic rule logalogb=log(ab)\log a - \log b = \log \left(\frac{a}{b}\right): log(8x2)=log32\log \left(\frac{8x}{2}\right) = \log 32 This simplifies to: log(4x)=log32\log (4x) = \log 32

  2. Since the logarithms are equal, we can drop the log and equate the inside of the logarithms: 4x=324x = 32

  3. Solve for xx: x=324=8x = \frac{32}{4} = 8

Thus, the solution is x=8x = 8.

Answer: x=8x = 8 (This matches one of the choices in the image you uploaded).

Would you like further explanation or help with similar problems? Here's some related practice to deepen your understanding:

  1. How do you handle equations involving logarithms with different bases?
  2. What if the equation involved logx+logy\log x + \log y instead of subtraction?
  3. Can you explain the logarithmic identity logab=bloga\log a^b = b \log a?
  4. How can you verify the solution to a logarithmic equation?
  5. What’s the domain of a logarithmic function?

Tip: Always remember to check that the argument of a logarithm is positive, as the logarithm of a non-positive number is undefined.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Equations
Properties of Logarithms

Formulas

log a - log b = log (a/b)
If log A = log B, then A = B

Theorems

Properties of Logarithms
Equality Property of Logarithms

Suitable Grade Level

Grades 9-12