Math Problem Statement

Express the equations 1024^(4/5) = 256 and 16^(-1/2) = 0.25 in logarithmic form.

Solution

Let’s solve the problems in the uploaded image.

Problem 7:

Given equation: 10244/5=2561024^{4/5} = 256

To express this in logarithmic form:

  • The general logarithmic form is logb(a)=c\log_b(a) = c if bc=ab^c = a.

Here, b=1024b = 1024, a=256a = 256, and c=45c = \frac{4}{5}.
Thus, the logarithmic form is: log1024(256)=45.\log_{1024}(256) = \frac{4}{5}.


Problem 8:

Given equation: 161/2=0.2516^{-1/2} = 0.25

To express this in logarithmic form:

  • Again, the general logarithmic form is logb(a)=c\log_b(a) = c if bc=ab^c = a.

Here, b=16b = 16, a=0.25a = 0.25, and c=12c = -\frac{1}{2}.
Thus, the logarithmic form is: log16(0.25)=12.\log_{16}(0.25) = -\frac{1}{2}.


Would you like further explanations or additional help? Here are 5 related questions to deepen understanding:

  1. How do you evaluate logarithms for non-standard bases (like base 1024 or 16)?
  2. How can logarithmic and exponential equations be used to solve real-world problems?
  3. What is the relationship between fractional exponents and roots?
  4. Can you explain how negative exponents change the outcome of a power equation?
  5. How do you verify logarithmic expressions through exponential equivalence?

Tip: When working with logarithmic forms, always double-check by converting back to the exponential equation to verify your result.

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Math Problem Analysis

Mathematical Concepts

Exponents
Logarithms
Logarithmic Form

Formulas

log_b(a) = c if b^c = a

Theorems

Logarithmic and Exponential Relationship

Suitable Grade Level

Grades 9-12