Problem 102 - College Algebra

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{5}(x+4)=2 $$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 21\).

Step 1: Convert to Exponential Form

The relationship between logarithmic and exponential forms is defined as follows: \(\log_b a = c\) is equivalent to \(b^c = a\). Therefore, the given equation \(\log_5(x+4) = 2\) can be rewritten in its equivalent exponential form as \(5^2 = x+4\).

Step 2: Simplify the Exponential Expression

Solving \(5^2\) yields \(25\). Therefore, the equation simplifies to \(25 = x + 4\).

Step 3: Solve for \(x\)

To isolate \(x\), subtract \(4\) from both sides of the equation, which results in \(21 = x\).

Key Concepts

Exponential Form

In mathematics, understanding the relationship between exponential form and logarithmic form is crucial, especially when solving equations. The exponential form of a number is a way of expressing its powers. In simple terms, if you have a base number raised to a certain power, it's written as \( b^c = a \), where \(b\) is the base and \(c\) is the exponent, resulting in \(a\).
For example, in the exercise where we have \( \log _{5}(x+4)=2 \), the exponential form is \( 5^2 = x+4 \). Here, we recognize that the base is 5, the exponent is 2, and it equals \(x+4\). This conversion from logarithm to exponential is key to simplifying and solving the equation.

• Always identify your base and the result of the logarithm.
• Convert it by raising the base to the power of the result to switch to exponential form.

This is the cornerstone for figuring out equations involving logarithms.

Logarithms

Logarithms can seem complex at first, but they are simply the inverse operation of exponentiation. If you know how exponents work, then understanding logarithms shouldn't be too difficult. The statement \( \log_b a = c \) means that \(b^c = a\). This implies that the logarithm \( \log_b a \) tells you the power to which the base \(b\) must be raised to get the number \(a\).
In the exercise provided, we have \( \log _{5}(x+4)=2 \), which implies \(5^2 = x+4\). Here:

• 5 is the base
• (x+4) is the result (a) the base is raised to
• 2 is the power

Understanding this relationship helps in transitioning between logarithmic and exponential forms, which is essential in solving these kinds of equations.

Solving Equations

Solving equations involves finding what value of the variable makes the equation true. Once you convert a logarithmic form to exponential, you can simplify and solve it efficiently.
In our specific problem, once we converted \( \log _{5}(x+4)=2 \) to the exponential form \( 5^2 = x+4 \), we computed \(5^2\) to get 25.

• This results in the equation \( 25 = x + 4 \).
• To solve for \(x\), isolate the variable by subtracting 4 from both sides: \( 25 - 4 = x \).
• This simplifies to \( x = 21 \).

By converting the problem to exponential form and logically simplifying step-by-step, you easily arrive at the solution. It is these step-by-step simplifications and transitions that aid in solving equations efficiently.