Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log(3x−3)=log(x+1)+log 4

Verified step by step guidance

1. Recall the logarithmic property that states: $\log \left(a\right)+\log \left(b\right)=\log (a\cdot b)$. Use this to combine the right side of the equation: $\log (x+1)+\log \left(4\right)=\log (4\cdot (x+1\left)\right)$.
2. Set the arguments of the logarithms equal to each other because if $\log (A)\; =\log (B)$, then $A\; =\; B$. So, write the equation: $3x\; -\; 3\; =\; 4(x\; +\; 1)$.
3. Expand the right side of the equation: $4(x\; +\; 1)\; =\; 4x\; +\; 4$. So the equation becomes $3x\; -\; 3\; =\; 4x\; +\; 4$.
4. Solve the linear equation for $x$: subtract $3x$ from both sides to get $-3\; =\; x\; +\; 4$, then subtract 4 from both sides to isolate $x$.
5. Check the solution by substituting the value of $x$ back into the original logarithmic expressions to ensure the arguments are positive (since the domain of $log$ requires positive arguments). Reject any solution that makes the argument of any logarithm non-positive.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule log(a) + log(b) = log(ab), is essential for combining or simplifying logarithmic expressions. This allows the equation to be rewritten in a simpler form, making it easier to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all input values for which the argument is positive. When solving logarithmic equations, it is crucial to check that solutions do not make any logarithm’s argument zero or negative, as these are not valid in the real number system.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, then converting to an exponential form or equating arguments when logs have the same base. This process leads to an algebraic equation that can be solved for the variable.