Math Problem Statement

Rewrite log_5(25) = 2 as an exponential equation.

Solution

The given problem asks you to rewrite the logarithmic equation log5(25)=2\log_5(25) = 2 as an exponential equation using the property from Theorem 6.2, which states that:

ba=cif and only iflogb(c)=ab^a = c \quad \text{if and only if} \quad \log_b(c) = a

Step-by-step solution:

You are given:

log5(25)=2\log_5(25) = 2

Using the property from the theorem, logb(c)=a\log_b(c) = a can be rewritten as ba=cb^a = c.

In this case:

  • b=5b = 5,
  • a=2a = 2,
  • c=25c = 25.

So, you can rewrite the logarithmic equation log5(25)=2\log_5(25) = 2 as the exponential equation:

52=255^2 = 25

This confirms that 52=255^2 = 25, so the rewritten exponential form is correct.

Would you like any further clarification on this, or need help with more examples?

Related questions:

  1. What is the exponential form of log3(9)=2\log_3(9) = 2?
  2. Can you convert log10(1000)=3\log_{10}(1000) = 3 into exponential form?
  3. How would you rewrite log7(49)=2\log_7(49) = 2 in exponential form?
  4. How do you express log2(8)=3\log_2(8) = 3 as an exponential equation?
  5. Can you explain why logb(1)=0\log_b(1) = 0 for any base bb?

Tip:

To convert a logarithmic equation into exponential form, remember that logb(c)=a\log_b(c) = a is equivalent to ba=cb^a = c, which can help simplify or solve many problems related to logarithms.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents

Formulas

log_b(c) = a is equivalent to b^a = c

Theorems

Theorem: b^a = c if and only if log_b(c) = a

Suitable Grade Level

Grades 8-10