Questions: Use properties of logarithms to find a value for x. 3 log (4x+7)-1=3 Round your answer to three decimal places. x=

Transcript text: Use properties of logarithms to find a value for $x$. \[ 3 \log (4 x+7)-1=3 \] Round your answer to three decimal places. \[ x= \] $\square$

Solution

Solution Steps

To solve the equation $3 \log (4x+7) - 1 = 3$, we can first isolate the logarithmic term by adding 1 to both sides. Then, divide by 3 to solve for the logarithm. Next, we exponentiate both sides to eliminate the logarithm and solve for $x$. Finally, we round the result to three decimal places.

Step 1: Isolate the Logarithmic Term

Starting with the equation: \[ 3 \log(4x + 7) - 1 = 3 \] we add 1 to both sides: \[ 3 \log(4x + 7) = 4 \]

Step 2: Divide by 3

Next, we divide both sides by 3: \[ \log(4x + 7) = \frac{4}{3} \]

Step 3: Exponentiate to Eliminate the Logarithm

We exponentiate both sides to eliminate the logarithm: \[ 4x + 7 = 10^{\frac{4}{3}} \]

Step 4: Solve for $x$

Now, we isolate $x$: \[ 4x = 10^{\frac{4}{3}} - 7 \] \[ x = \frac{10^{\frac{4}{3}} - 7}{4} \]

Calculating $10^{\frac{4}{3}}$ gives approximately $21.5443$. Thus: \[ x = \frac{21.5443 - 7}{4} = \frac{14.5443}{4} \approx 3.6361 \]