Problem 65 - Algebra A Combined Function

Question

Solve. $$ 3^{\log _{3} 5}=x $$

Step-by-Step Solution

Answer

The solution is x = 5.

Step 1: Understand the logarithmic identity

The expression \( 3^{\log_{3} 5} \) uses a logarithmic identity where \( a^{\log_{a} b} = b \). This is a special property of logarithms and exponentials that helps simplify such expressions directly.

Step 2: Apply the logarithmic identity

Given the identity \( a^{\log_{a} b} = b \), apply it to the expression \( 3^{\log_{3} 5} \). Here, \( a = 3 \) and \( b = 5 \). Therefore, \( 3^{\log_{3} 5} = 5 \).

Step 3: Identify the solution

From the application of the logarithmic identity in the previous step, we see that the value of \( x \) in the expression \( 3^{\log_{3} 5} = x \) is \( x = 5 \).

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of mathematics featuring an equation of the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent. These functions are very powerful and have numerous applications in science and engineering. One important characteristic of exponential functions is that they involve a constant base raised to a variable exponent.

• Growth or Decay: Exponential functions characterize growth or decay processes such as population growth or radioactive decay. The constant base \( a \) determines whether the function is increasing (if \( a > 1 \)) or decreasing (if \( 0 < a < 1 \)).
• Rate of Change: In exponential growth, the rate of change is proportional to the value of the function itself. This means the larger the value, the faster the rate of growth.

Understanding exponential functions allows us to decipher more complicated mathematical concepts, including those involving logarithms as inverse operations.

Simplifying Expressions

Simplifying expressions lets mathematicians reduce a complex expression into a more manageable form. Various strategies can be applied, making solving equations simpler and solutions easier to find. For example, recognizing patterns and applying identities can vastly simplify the solving process.

• Using Identities: A useful identity in logarithmic and exponential expressions is \( a^{\log_{a} b} = b \). This allows simplification of complex-looking expressions directly.
• Elimination Techniques: Factorizing elements and canceling like terms help reduce the components of an expression. This makes the expression clearer.

Simplifying allows you to achieve a final expression or solution with less complexity and avoid unnecessary calculations.

Logarithms

Logarithms are the inverses of exponential functions and are essential in solving exponential equations. They answer the question, "What power do we raise a specific number to, to get another number?" The general form of a logarithm is \( \log_a b \), meaning the power you raise \( a \) to, to get \( b \).

Properties and Identities

• Inverse Property: Since logarithms are inverses of exponentials, \( a^{\log_a b} \) simplifies to \( b \). This property is exceedingly useful in solving logarithmic equations.
• Logarithmic Identities: Logarithms follow specific rules such as the product, quotient, and power rules, which simplify expressions like multiplying or dividing logarithmic values, or raising a logarithm to a power.

Understanding how logarithms operate and their properties can help demystify complex expressions and equations, making them easier to solve.