Questions: Solve the following equation: log5(6x+3)=2 Write your answers exactly (i.e. no decimal approximations and simplified if possible), separated by a comma. x= Write your answers as decimals rounded to four decimal places. x ≈

Solve the following equation:
log5(6x+3)=2

Write your answers exactly (i.e. no decimal approximations and simplified if possible), separated by a comma.
x=

Write your answers as decimals rounded to four decimal places.
x ≈

Transcript text: Solve the following equation: \[ \log _{5}(6 x+3)=2 \] Write your answers exactly (i.e. no decimal approximations and simplified if possible), separated by a comma. \[ x= \] Write your answers as decimals rounded to four decimal places. \[ x \approx \]

Solution

Solution Steps

To solve the equation \(\log _{5}(6x+3)=2\), we need to convert the logarithmic equation into an exponential form. This will allow us to solve for \(x\) algebraically. Once we have the exact solution, we can then calculate the decimal approximation.

Step 1: Convert the Logarithmic Equation

We start with the equation: \[ \log_{5}(6x + 3) = 2 \] To convert this logarithmic equation into exponential form, we rewrite it as: \[ 6x + 3 = 5^2 \]

Step 2: Simplify the Exponential Expression

Calculating \(5^2\) gives us: \[ 6x + 3 = 25 \]

Step 3: Solve for \(x\)

Next, we isolate \(x\) by subtracting 3 from both sides: \[ 6x = 25 - 3 \] This simplifies to: \[ 6x = 22 \] Now, dividing both sides by 6 yields: \[ x = \frac{22}{6} = \frac{11}{3} \]

Step 4: Calculate the Decimal Approximation

To find the decimal approximation of \(x\), we compute: \[ x \approx 3.6667 \]