Use the laws of logarithms to solve log_(2)(16 x)+log_(2)(x+1)=3+log_(2)(x+6)

Combine logs using product property: Combine the logs on the left side using the product property of logarithms; log_b(m) + log_b(n) = log_b(m*n). Subtract to isolate logarithmic terms: Subtract log_(2)(x+6) from both sides to isolate the logarithmic terms on one side. Apply quotient property of logarithms: Apply the quotient property of logarithms; log_b(m) - log_b(n) = log_b(m/n). Convert to exponential equation: Convert the logarithmic equation to an exponential equation; if log_b(m) = n, then b^n = m. Simplify exponential expression: Simplify the exponential expression. Eliminate denominator by multiplying: Multiply both sides by (x+6) to eliminate the denominator. Distribute and expand equation: Distribute and expand both sides of the equation. Move terms to set quadratic equation to zero: Move all terms to one side to set the quadratic equation to zero. Combine like terms: Combine like terms. Divide entire equation by 8: Divide the entire equation by 8 to simplify. Factor quadratic equation: Factor the quadratic equation. Set factors equal to zero: Set each factor equal to zero and solve for x. Solve first equation for x: Solve the first equation for x. Solve second equation for x: Solve the second equation for x.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Use the laws of logarithms to solve $\newline$ $\log_{2}(16x)+\log_{2}(x+1)=3+\log_{2}(x+6)$

View step-by-step help

Home

Math Problems

Precalculus

Product property of logarithms

Full solution

Q. Use the laws of logarithms to solve

\newline

\log_{2}(16x)+\log_{2}(x+1)=3+\log_{2}(x+6)

Combine logs using product property: Combine the logs on the left side using the product property of logarithms; $\log_b(m) + \log_b(n) = \log_b(m*n)$ .
Subtract to isolate logarithmic terms: Subtract $\log_{2}(x+6)$ from both sides to isolate the logarithmic terms on one side.
Apply quotient property of logarithms: Apply the quotient property of logarithms; $\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)$ .
Convert to exponential equation: Convert the logarithmic equation to an exponential equation; if $\log_b(m) = n$ , then $b^n = m$ .
Simplify exponential expression: Simplify the exponential expression.
Eliminate denominator by multiplying: Multiply both sides by $(x+6)$ to eliminate the denominator.
Distribute and expand equation: Distribute and expand both sides of the equation.
Move terms to set quadratic equation to zero: Move all terms to one side to set the quadratic equation to $0$ .
Combine like terms: Combine like terms.
Divide entire equation by $8$ : Divide the entire equation by $8$ to simplify.
Factor quadratic equation: Factor the quadratic equation.
Set factors equal to zero: Set each factor equal to $0$ and solve for $x$ .
Solve first equation for $x$ : Solve the first equation for $x$ .
Solve second equation for $x$ : Solve the second equation for $x$ .