Math Problem Statement

log equations, log form from exponential

Solution

To convert between logarithmic and exponential forms:

  1. Logarithmic Form to Exponential Form: logb(y)=xbx=y\log_b(y) = x \quad \Rightarrow \quad b^x = y

    • Here, bb is the base, xx is the exponent, and yy is the result.
  2. Exponential Form to Logarithmic Form: bx=ylogb(y)=xb^x = y \quad \Rightarrow \quad \log_b(y) = x

Examples:

Example 1:

Logarithmic Form: log2(8)=3\log_2(8) = 3 Convert to Exponential Form: 23=82^3 = 8

Example 2:

Exponential Form: 102=10010^2 = 100 Convert to Logarithmic Form: log10(100)=2\log_{10}(100) = 2

Properties of Logarithms:

  1. logb(1)=0\log_b(1) = 0 because b0=1b^0 = 1.
  2. logb(b)=1\log_b(b) = 1 because b1=bb^1 = b.
  3. logb(xn)=nlogb(x)\log_b(x^n) = n \log_b(x).

Would you like detailed explanations or further examples?


Related Questions:

  1. How do logarithmic equations differ from exponential equations?
  2. What are the common bases used in logarithms?
  3. How do you solve equations involving logarithms?
  4. What are the laws of logarithms (product, quotient, power)?
  5. How is the natural logarithm (ln\ln) different from other logarithms?

Tip: Remember, the base of a logarithm must always be positive and cannot equal 1.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponential Functions
Algebra

Formulas

log_b(y) = x ↔ b^x = y
log_b(x^n) = n log_b(x)
log_b(1) = 0
log_b(b) = 1

Theorems

Properties of Logarithms
Definition of Logarithms as Inverse of Exponentials

Suitable Grade Level

Grades 8-10