Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 4^{x}-10 \cdot 4^{-x}=3 $$

Short answer

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

Step-by-step solution

- Rewrite the equation using a single base

Given the equation: \[ 4^{x} - 10 \cdot 4^{-x} = 3 \]Rewrite the term involving the negative exponent using a positive exponent: \[ 4^{x} - 10 \cdot \frac{1}{4^{x}} = 3 \]

- Substitution

Let \( y = 4^x \). So, \( \frac{1}{y} = 4^{-x} \).Substitute \( y \) into the equation: \[ y - 10 \cdot \frac{1}{y} = 3 \]

- Clear the fraction

Multiply both sides by \( y \): \[ y^2 - 10 = 3y \]

- Form a quadratic equation

Rearrange the equation into the standard quadratic form: \[ y^2 - 3y - 10 = 0 \]

- Solve the quadratic equation

Factor the quadratic equation: \[ (y - 5)(y + 2) = 0 \]So, the solutions are: \[ y = 5 \]\[ y = -2 \]

- Back-substitute and find \( x \)

Recall \( y = 4^x \).For \( y = 5 \): \[ 4^x = 5 \]Take the logarithm of both sides: \[ x \cdot \log(4) = \log(5) \]Solve for \( x \): \[ x = \frac{\log(5)}{\log(4)} \]For \( y = -2 \): \[ 4^x = -2 \]There is no real solution since the exponential function cannot be negative.

Key concepts

Exponential Functions

Exponential functions are essential in mathematics due to their unique properties. An exponential function is defined as \( f(x) = a \times b^x \), where \( a \) and \( b \) are constants, and \( b \) is the base. In these functions, the variable, \( x \), is in the exponent. This allows for rapid growth or decay rates, unlike linear functions which grow at a constant rate.

For example, in the equation \( 4^x - 10 \times 4^{-x} = 3 \), the terms \( 4^x \) and \( 4^{-x} \) are exponential. Measuring quantities like population growth, radioactive decay, and compound interest often use exponential functions.

Understanding the behavior of exponential functions:

• The larger the base, the steeper the increase or decrease.
• When the exponent is positive, the function depicts growth. A negative exponent describes decay.
• Exponential functions never touch the x-axis; they get infinitely close but do not reach zero.

An important characteristic is that these functions are one-to-one, meaning each input has a specific output, vital for solving equations like the one in our exercise.

Quadratic Equations

Solving quadratic equations often plays a crucial role in various mathematical problems. A quadratic equation has the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are coefficients. In our exercise, after substitution, an exponential equation is transformed into a quadratic equation.

To solve \( y^2 - 3y - 10 = 0 \), follow these steps:

• Factorize the quadratic. In this case, it factors into \( (y - 5)(y + 2) = 0 \). This method works when you can easily identify factors of \( c \) that add up to \( b \).
• Set each factor equal to zero. This gives us \( y - 5 = 0 \) or \( y + 2 = 0 \).
• Solve for \( y \). Hence, we find \( y = 5 \) or \( y = -2 \).

Quadratic equations frequently appear in various contexts such as area problems, projectile motion, and optimization problems. Recognizing and solving quadratics is fundamental in algebra and other areas of mathematics.

Logarithms

Logarithms are the inverse functions of exponential functions. They help in solving equations where the unknown is in the exponent, making them a powerful tool in various fields. The logarithm of a number is the exponent to which the base must be raised to obtain that number. Specifically, for \( b^x = y \), \( \text{log}_b(y) = x \).

In our exercise, we solve for \( x \) using logarithms. When we have \( 4^x = 5 \), we take the logarithm of both sides:

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

Solving for \( x \) gives:

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

Logarithm properties helpful in solving these equations include:

• \( \text{log}(ab) = \text{log}(a) + \text{log}(b) \)
• \( \text{log}\frac{a}{b} = \text{log}(a) - \text{log}(b) \)
• \( \text{log}(a^b) = b \times \text{log}(a) \)

Logarithms convert multiplicative processes into additive ones, making it easier to handle complex calculations. They are widely used in sciences like chemistry and physics to manage exponential growth, such as pH in solutions or sound intensity in decibels.