Problem 25 - Algebra A Combined Function

Question

Solve. $$ 32^{2 x-3}=2 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{8}{5} \).

Step 1: Equate Bases

First, we notice that the left side of the equation, \( 32^{2x-3} \), has a base of 32, while the right side is simply 2. Since 32 can be expressed as a power of 2, we rewrite 32 as \( 2^5 \). This gives us \((2^5)^{2x-3} = 2 \).

Step 2: Simplify the Exponentiation

Next, simplify by applying the power of a power property, \( (a^m)^n = a^{mn} \). This means \((2^5)^{2x-3} \) becomes \( 2^{5(2x-3)} \). Thus, the equation is \( 2^{5(2x-3)} = 2^1 \).

Step 3: Set the Exponents Equal

Since the bases on both sides are now the same, we set the exponents equal to each other: \[ 5(2x-3) = 1 \]

Step 4: Solve for x

First, expand the left side: \[ 5(2x-3) = 10x - 15 \] Now, equate and solve for \( x \):\[ 10x - 15 = 1 \]\[ 10x = 16 \]\[ x = \frac{16}{10} = \frac{8}{5} \].

Key Concepts

Logarithmic Functions

When solving exponential equations, understanding logarithmic functions can be incredibly helpful. A logarithmic function is essentially the inverse of an exponential function. This means if you know how to work with one, you can solve problems related to the other. For example, if the equation is of the form \( b^y = x \), a logarithm can help you find \( y \). This is done using the expression \( \log_b(x) = y \), which reads as "the logarithm of \( x \) with base \( b \) equals \( y \)."
To better understand:

• \( \log(100) = 2 \) since \( 10^2 = 100 \)
• \( \log_2(8) = 3 \) because \( 2^3 = 8 \)

You get the idea—it shifts your thinking from multiplication to addition, which can make complex equations seem much simpler to solve. If you ever face an exponential equation that seems daunting, remember that logarithms can help you break it down.

Power of a Power Property

Understanding properties of exponents like the power of a power property can drastically simplify solving exponential equations. This property states that \((a^m)^n = a^{m \cdot n}\). What does this mean in simpler terms? When you have an exponent raised to another exponent, you can multiply the exponents together.
This property is particularly useful when you want to equalize the bases on both sides of an equation. For instance, if we start with \((2^5)^{2x-3}\), applying this property simplifies it into \(2^{5(2x-3)}\), which is much easier to handle.

• It streamlines complex exponential equations.
• You can combine multiple exponential expressions effectively.

By using the power of a power property, you take a complicated element and make it manageable, which is key in solving equations efficiently.

Solving Equations with Exponents

Learning to solve equations with exponents involves a systematic process. Usually, it starts with making the bases on both sides of the equation the same. This enables you to set the exponents equal to each other and solve for the unknown.

• Step 1: Make the bases the same if possible. For example, transforming \(32\) into \(2^5\).
• Step 2: Apply properties of exponents, like the power of a power property, to simplify if necessary.
• Step 3: Set equivalent exponents equal to each other. For the equation \(32^{2x-3} = 2\), this becomes \(5(2x-3) = 1\) once simplified.
• Step 4: Solve for \(x\) by expanding and isolating the variable.

Each step is designed to simplify the equation gradually, making it possible to isolate the variable you're solving for. Practicing these steps will give you skills that apply not just to specific problems but to solving a wide range of exponential equations.