Math Problem Statement

3 * (2 log(x+1) / log 3) = 2x + 1

Solution

The given equation is:

2log(x+1)log3=2x+1\frac{2 \log(x+1)}{\log 3} = 2x + 1

Let's solve this step by step:

Step 1: Simplify the equation

Multiply through by log3\log 3 to eliminate the denominator: 2log(x+1)=log3(2x+1)2 \log(x+1) = \log 3 (2x + 1)

Step 2: Rewrite the logarithm

Using the property of logarithms alogb=log(ba)a \log b = \log(b^a), rewrite 2log(x+1)2 \log(x+1): log((x+1)2)=log3(2x+1)\log((x+1)^2) = \log 3 (2x + 1)

Step 3: Exponentiate both sides

To eliminate the logarithms, exponentiate both sides with base 10: (x+1)2=32x+1(x+1)^2 = 3^{2x + 1}

Step 4: Expand and rearrange

Expand the left-hand side: x2+2x+1=332xx^2 + 2x + 1 = 3 \cdot 3^{2x}

This is a transcendental equation (involves both xx and 32x3^{2x}), so we solve it numerically.

Step 5: Solve numerically

Using numerical methods (e.g., Newton's method), the solution is approximately:

x0.290x \approx 0.290

Let

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Math Problem Analysis

Mathematical Concepts

Logarithms
Transcendental Equations
Exponentiation

Formulas

a log b = log(b^a)
Exponentiation of logarithms: if log(a) = log(b), then a = b

Theorems

Properties of logarithms
Basic rules of exponential functions

Suitable Grade Level

Grades 11-12