Question

$$ \text { Solve: } \log _{3} 9^{2 x+3}=x^{2}+1 $$

Short answer

x = 5 or x = -1

Step-by-step solution

- Simplify the logarithmic expression

Rewrite the term inside the logarithm. Note that \( 9 = 3^2 \). Therefore, \( 9^{2x+3} = (3^2)^{2x+3} \). Using the power of a power rule, we get \( (3^2)^{2x+3} = 3^{4x+6} \). So, the equation becomes \(\log_{3}(3^{4x+6})=x^2+1\).

- Apply the logarithm properties

Use the property of logarithms that states: \( \log_{a}(a^b) = b \). Applying this, we get \( 4x + 6 = x^2 + 1 \).

- Form a quadratic equation

Rearrange the equation to set it to zero: \( x^2 - 4x + 1 - 6 = 0 \). This simplifies to \( x^2 - 4x - 5 = 0 \).

- Solve the quadratic equation

Factorize the quadratic equation. We need two numbers that multiply to -5 and add up to -4, which are -5 and 1. Hence, \( (x - 5)(x + 1) = 0 \).

- Find the solutions

Set each factor to zero and solve for x: \( x - 5 = 0 \) gives \( x = 5 \) and \( x + 1 = 0 \) gives \( x = -1 \). Therefore, the solutions are \( x = 5 \) and \( x = -1 \).

Key concepts

Logarithm Properties

Logarithms are a crucial concept in mathematics, helping us to solve equations involving exponential functions. One essential property of logarithms used to solve the given problem is the exponent rule: \( \log_{a}(a^b) = b\). This property simplifies the logarithm by cancelling out the base and the power. For example, in \( \log_{3}(3^{4x+6}) = 4x + 6 \), the base 3 is cancelled by the logarithm, making the equation much easier to solve. Another key property is the change of base formula, but it isn't used in this problem.

Power of a Power Rule

The power of a power rule is an exponent principle that states \( (a^m)^n = a^{mn} \). This rule allows us to simplify expressions involving multiple exponents. In our problem, \( 9^{2x+3} \) can be written as \( (3^2)^{2x+3} \). Applying the power of a power rule, we get \( 3^{4x+6} \.\). This transformation is necessary to match the base of the logarithm and apply the logarithm property mentioned earlier.

Quadratic Equation

Solving the quadratic equation is a common step in algebra. A quadratic equation takes the form \( ax^2 + bx + c = 0 \.\). In the problem, after applying the logarithm property, we are left with \( x^2 - 4x - 5 = 0 \.\). To solve it, we must find values of x that satisfy this equation. Quadratic equations can be solved using different methods such as factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).

Factorization

Factorization is a method used to solve equations by expressing a polynomial as a product of simpler polynomials. In the given problem, the quadratic equation \( x^2 - 4x - 5 = 0 \) can be factorized to \( (x - 5)(x + 1) = 0 \.\). To factorize, we seek two numbers that multiply to the constant term (-5) and add up to the coefficient of the linear term (-4). These numbers are -5 and 1. Setting each factor to zero gives the solutions: \( x - 5 = 0 \) leads to \( x = 5 \), and \( x + 1 = 0 \) leads to \( x = -1 \).