Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 25^{x}-8 \cdot 5^{x}=-16 $$

Short answer

x = \( \frac{\text{log}(4)}{\text{log}(5)} \)

Step-by-step solution

Rewrite Exponential Terms

Rewrite the equation to have the same base. Observe that \(25\) can be written as \(5^2\). So the equation becomes: \[ (5^2)^{x} - 8 \times 5^{x} = -16 \] Which simplifies to: \[ 5^{2x} - 8 \times 5^{x} = -16 \]

Substitute Variable

Let \( y = 5^{x} \). This simplifies the equation to: \[ y^2 - 8y = -16 \]

Set Equation to Zero and Factor

Rewrite the equation so that one side is zero: \[ y^2 - 8y + 16 = 0 \] Now factor the quadratic equation: \[ (y - 4)^2 = 0 \]

Solve for y

Solve for \( y \) by taking the square root of both sides: \[ y - 4 = 0 \] Thus, \[ y = 4 \]

Back Substitute y

Recall that \( y = 5^{x} \). Replace \( y \) with \( 5^{x} \) in the equation: \[ 5^{x} = 4 \]

Solve for x

Using logarithms, solve for \( x \): \[ x = \frac{\text{log}(4)}{\text{log}(5)} \]

Key concepts

solving exponential equations

Solving exponential equations often seems tricky because of the exponents involved, but it can be made easier by following a clear step-by-step approach. Let's look at the exercise example where we need to solve the equation:
\( 25^{x}-8 \cdot 5^{x}=-16 \).
To begin solving exponential equations:

• Rewrite each term to have the same base.

The original problem gives us a hint because 25 can be written as \(5^2\). After rewriting, the equation becomes \[ (5^2)^{x} - 8 \times 5^{x} = -16 \]
This simplifies to \[ 5^{2x} - 8 \times 5^{x} = -16 \]
When the bases are the same, you can focus on simplifying the exponent terms to solve the equation.

substitution method

The substitution method can simplify complex exponential equations. In our exercise, we substitute to reduce the complexity:
Let \( y = 5^{x} \), which transforms the exponential equation into a quadratic form:
\[ y^2 - 8y = -16 \]

To use the substitution method effectively:

• Identify a substitution that simplifies the equation.
• Replace the complex term with a simpler variable.

Now, our equation becomes a standard quadratic equation that can be set to zero and factored:
\[ y^2 - 8y + 16 = 0 \]
Factoring the quadratic equation gives us:
\[ (y - 4)^2 = 0 \]
Solving this, we get:
\[ y - 4 = 0 \Rightarrow y = 4 \]
Finally, we replace \( y = 5^x \) back into the equation to find the original variable.

logarithms

Logarithms are essential when solving exponential equations, especially when we need to find the value of exponents. After substitution and solving for y in the exercise:
\[ y = 4 \]
We remember that \( y = 5^x \), so:
\[ 5^x = 4 \]
To solve for \( x \), we use logarithms:

• Apply the logarithm to both sides of the equation.
• Use the properties of logarithms to isolate the variable \( x \).

Applying logarithms, we get:
\[ x = \frac{{\text{log}(4)}}{{\text{log}(5)}} \]
This isolates \( x \) and allows us to find the exact solution even if it is irrational.
This method is powerful for solving equations where the exponents are not easily simplified.