Question

Solve each logarithmic equation. Express irrational solutions in exact form. $$ \log _{3}(3 x-1)=2 $$

Short answer

The solution is \(x = \frac{10}{3}\).

Step-by-step solution

Understand the Problem

The given equation is \(\log _{3}(3 x-1)=2\). We need to solve for \(x\).

Rewrite the Logarithmic Equation in Exponential Form

Using the property \(\log_b(a) = c \Longleftrightarrow b^c = a\), rewrite the logarithmic equation \(\log _{3}(3 x-1)=2\) in exponential form. This gives us \[ 3^2 = 3x - 1 \]

Solve the Exponential Equation

We now have the equation \(3^2 = 3x - 1\). Simplify the left side: \ 3^2 = 9 \ so the equation becomes \[ 9 = 3x - 1 \]

Isolate the Variable

To solve for \(x\), first add 1 to both sides of the equation: \[ 9 + 1 = 3x \] Simplify: \[ 10 = 3x \]

Solve for x

Divide both sides by 3 to isolate \(x\): \[ x = \frac{10}{3} \]

Key concepts

exponential form

To solve a logarithmic equation, one powerful method is to rewrite it in exponential form. This leverages the fundamental relationship between logarithms and exponents. The equation \(\text{log}_{b}(a) = c\) implies that \[ b^c = a \]. This conversion is essential to simplifying and solving the equation.
For example, given \(\text{log}_{3}(3x - 1) = 2\), we use the property mentioned to rewrite it. This becomes \[ 3^2 = 3x - 1 \].
When rewritten, logarithmic functions often become simpler to manage. Understanding this relationship helps us see the problem from a different perspective and find a clearer path to the solution.

isolate the variable

Once we have the equation in exponential form, our next goal is to isolate the variable. This involves rearranging the equation so that the variable is by itself on one side.
Starting with the equation \(\text{log}_{3}(3x - 1) = 2\) and rewriting it in exponential form gives us \[ 9 = 3x - 1 \].
To isolate the variable, add 1 to both sides: \[ 9 + 1 = 3x \]
Then, simplify to \[ 10 = 3x \].
By manipulating the equation step-by-step, we bring the unknown variable to one side, making the problem more straightforward to solve.

simplify

After isolating the variable, the final step is to simplify the equation. This typically means performing arithmetic operations to solve for the variable explicitly.
Continuing from our isolated form \[ 10 = 3x \], we divide both sides by 3: \[ x = \frac{10}{3} \].
By performing this division, we find the value of the variable. Simplifying an equation is crucial in obtaining the final answer in its simplest form.
Understanding how to simplify ensures that we accurately and efficiently find solutions to the problem at hand.