Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(5 \log _{3}(x+1)=12\)

Short answer

The approximate solution to the given logarithmic equation is \(x \approx 9.759\).

Step-by-step solution

Isolate the Logarithmic Term

To begin the solution, isolate the logarithmic term. This can be done by dividing both sides of the equation by 5. This gives: \(\log_{3}(x+1) = \frac{12}{5}\).

Convert to Exponential Form

The next step is to convert the logarithmic equation to an exponential equation. This gives: \(3^{\frac{12}{5}} = x + 1\).

Subtract 1 from Both Sides

Now, we isolate 'x' by subtracting 1 from both sides of the equation. This will provide the exact value of 'x'. This gives: \(x = 3^{\frac{12}{5}} - 1\).

Approximation of Result

Finally, we will find the approximate value of 'x' by using a calculator, remembering to round to three decimal places. Calculating this yields: \(x \approx 9.759\).

Key concepts

Transforming Logarithmic to Exponential Form

Converting a logarithmic equation to its exponential form is a key step in solving logarithmic equations. This transformation is based on the fundamental definition of logarithms. Let's break it down: The general form of a logarithm is \( \log_{b}(a) = c \), which is equivalent to the exponential form \( b^{c} = a \). To convert \( \log_{3}(x+1) = \frac{12}{5} \) to exponential form, we use this principle. We raise the base \( 3 \) to the power \( \frac{12}{5} \) which then equals \( x+1 \) because the base \(3 \) in the logarithm indicates that it is the very same base we use to raise our exponent. So the equation becomes \( 3^{\frac{12}{5}} = x + 1 \).

Understanding this step is essential for moving forward with solving the equation. Remembering this correlation between logarithmic and exponential form can immensely simplify the process and is a foundation for grasping more complex logarithmic calculations.

Isolating the Logarithmic Term

When solving for a variable in a logarithmic equation, one of the first objectives is to isolate the logarithmic term. To tackle this, any constants or coefficients attached to the logarithmic term need to be managed. In our example, the initial equation is \( 5 \log _{3}(x+1)=12 \). The logarithmic term here is \( \log _{3}(x+1) \) and it is multiplied by 5.

The isolation process involves doing the inverse operation of what is being done to the logarithmic term. Since \( \log _{3}(x+1) \) is multiplied by 5, we divide both sides by 5 to isolate the logarithm, resulting in \( \log_{3}(x+1) = \frac{12}{5} \).

Isolating the logarithm makes it easier to apply the properties of logarithms and to ultimately solve for the variable. It’s a crucial step that sets up the stage for the remaining part of problem-solving.

Approximation of Logarithmic Equations

After isolating the logarithmic term and converting the equation to exponential form, the last step is to approximate the solution to a desired degree of accuracy. For logarithmic equations, this often requires the use of a calculator since the equations usually do not result in integer values.

In the example, after we've transformed and rearranged the equation to \( x = 3^{\frac{12}{5}} - 1 \), we are left with a base and an exponent that are not easily simplified without calculating tools. We enter this expression into a calculator to find the value of 'x'. However, since logarithmic values can be intricate, rounding to a specified number of decimal places is necessary.

In this case, the result must be rounded to three decimal places, giving us \( x \approx 9.759 \). This rounding makes the solution usable and assessable in real-life situations, where perfect accuracy is not always possible or required. When approximating, ensure that the rounding protocol follows mathematical standards, cutting at the correct decimal and adjusting the last digit when necessary.