Math Problem Statement

Solve each equation for x by applying properties of exponents and logarithms. log3(3x+6)/log3(81) = 1.

Solution

To solve this equation:

log3(3x+6)log3(81)=1,\frac{\log_3(3x + 6)}{\log_3(81)} = 1,

we apply the properties of logarithms step by step:


Step 1: Simplify log3(81)\log_3(81)

We know that 81=3481 = 3^4, so:

log3(81)=4.\log_3(81) = 4.

The equation now becomes:

log3(3x+6)4=1.\frac{\log_3(3x + 6)}{4} = 1.


Step 2: Eliminate the denominator

Multiply through by 4 to eliminate the denominator:

log3(3x+6)=4.\log_3(3x + 6) = 4.


Step 3: Rewrite in exponential form

Using the property logb(a)=c    bc=a\log_b(a) = c \implies b^c = a, rewrite the logarithmic equation:

34=3x+6.3^4 = 3x + 6.

Simplify 343^4:

81=3x+6.81 = 3x + 6.


Step 4: Solve for xx

Subtract 6 from both sides:

75=3x.75 = 3x.

Divide by 3:

x=25.x = 25.


Final Answer:

x=25x = 25

Let me know if you'd like further clarification!


Related Questions:

  1. What are the general properties of logarithms used here?
  2. How do you rewrite logarithmic equations in exponential form?
  3. Why is log3(81)=4\log_3(81) = 4? Explain step-by-step.
  4. Can the solution x=25x = 25 be verified? If so, how?
  5. What would the equation look like if log3(3x+6)=5\log_3(3x + 6) = 5?

Tip:

Always check your solutions by substituting back into the original equation to verify correctness.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logarithms
Exponential Equations
Properties of Logarithms

Formulas

log_b(a) = c implies b^c = a
log_b(b^n) = n

Theorems

Change of Base Property for Logarithms
Exponential-Logarithmic Relationship

Suitable Grade Level

Grades 9-11