What is the solution of the equation x log10 (10/3) + log10 3

• Logarithms

What is the solution of the equation
x log₁₀ (10/3) + log₁₀ 3 = log₁₀ (2 + 3^x) + x?

1. 10
2. 3
3. 1
4. 0

Answer

x log₁₀ (10/3) + log₁₀ 3 = log₁₀ (2 + 3^x) + x

x log₁₀ 10 - x log₁₀ 3 + log₁₀ 3 = log₁₀ (2 + 3^x) + x

x - x log₁₀ 3 + log₁₀ 3 = log₁₀ (2 + 3^x) + x

log₁₀ 3 - log₁₀ 3^x = log₁₀ (2 + 3^x)

log₁₀ (3/3^x) = log₁₀ (2 + 3^x)

3/3^x = 2 + 3^x

Let 3^x = t

3/t = 2 + t

t² +2t - 3 = 0

t = -3, 1

3^x = 1

x = 0

The correct option is D.