Question

Solve the equation. \(\log _5(5 x+10)=4\)

Short answer

The solution to the equation \(\log _5(5 x+10)=4\) is \(x = 123\).

Step-by-step solution

Convert logarithmic equation to exponential equation

Using the base \(b=5\), argument \(y=5x+10\), and result \(x=4\) from the logarithmic equation, we rewrite the equation in exponential form as: \(b^x = y\), which gives us \(5^4 = 5x+10\)

Simplify the equation

Calculate \(5^4\) to get \(625\), so our equation becomes \(625 = 5x+10\)

Isolate variable \(x\)

First, subtract 10 from both sides to isolate the term with \(x\): \(625 - 10 = 5x\), which simplifies to \(615 = 5x\). Then, divide both sides by 5 to solve for \(x\): \(x = 615/5\)

Final Simplification

After division, we find that \(x = 123\)

Key concepts

Exponential Form

Logarithms and exponentials are closely related. Understanding this relationship helps in converting a logarithmic equation into an exponential one, which often leads to simpler problem-solving. The logarithmic equation you start with is \( \log _5(5x+10)=4 \). In this context, the logarithm explains how many times the base \(5\) needs to be multiplied by itself to equal the given result.

To convert this to an exponential form, we use the parts of the logarithmic equation:

• **Base** \(b = 5\)
• **Argument** \(y = 5x + 10\)
• **Result** which is equivalent to \(x = 4\)

According to the properties of logarithms, you can express \( \log_b(y) = x \) as \( b^x = y \). Applying this to our equation, it becomes \( 5^4 = 5x + 10 \), taking us from a logarithmic representation into an exponential equation ready to be solved.

Solving Logarithmic Equations

Once the logarithmic equation is converted to an exponential form, solving it becomes straightforward. With the example, you've rewritten it to \( 5^4 = 5x + 10 \). The next steps involve simplifying this equation to make solving easier.

Begin by calculating \( 5^4 \). That means multiplying \(5\) by itself four times:

• \(5 \times 5 = 25\)
• \(25 \times 5 = 125\)
• \(125 \times 5 = 625\)

Now, replace \(5^4\) in the equation with \(625\), resulting in \( 625 = 5x + 10 \). This algebraic form is much simpler to handle, setting the stage for focusing solely on isolating the variable \(x\). This method highlights the power of breaking problems into smaller parts, helping to understand the logic and simplify the work.

Isolating Variables

The ultimate goal in these equations is to find the value of the variable that satisfies the equation. In this case, isolating \(x\) involves a few easy algebraic steps once you have \( 625 = 5x + 10 \).

To isolate \( x \), follow these actions:

• Subtract \(10\) from both sides to remove the constant term from the side with the variable:
• \( 625 - 10 = 5x \), simplifying to \( 615 = 5x \)
• This operation effectively narrows the problem down to a basic equation \( 5x \).

Next, divide both sides by the coefficient in front of \(x\), which is \(5\):

• \( \frac{615}{5} = x \)
• Calculate \( 615 / 5 \) to get \( x = 123 \)

These steps emphasize the ease with which a seemingly complex logarithmic equation can be demystified, clarifying the process of isolating variables in equation-solving tasks.