Solve $\log (7 x+7)=1$. Round to the nearest thousandth if necessary.
Answer (1)
Rewrite the logarithmic equation in exponential form: 7 x + 7 = 1 0 1 .
Simplify the equation: 7 x + 7 = 10 .
Subtract 7 from both sides and divide by 7: x = 7 3 .
Approximate the value of x to the nearest thousandth: 0.429 .
Explanation
Understanding the Problem We are given the equation lo g ( 7 x + 7 ) = 1 . The base of the logarithm is assumed to be 10. We need to solve for x and round to the nearest thousandth if necessary.
Converting to Exponential Form To solve the equation, we first rewrite the logarithmic equation in exponential form. Since the base of the logarithm is 10, we have 7 x + 7 = 1 0 1 .
Simplifying the Equation Now, we simplify the equation: 7 x + 7 = 10 .
Isolating the Term with x Next, we subtract 7 from both sides of the equation: 7 x = 10 − 7 , which simplifies to 7 x = 3 .
Solving for x Now, we divide both sides by 7 to solve for x : x = 7 3 .
Approximating the Solution Finally, we approximate the value of x to the nearest thousandth. 7 3 ≈ 0.42857 . Rounding to the nearest thousandth, we get x ≈ 0.429 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and modeling population growth. For example, if we know the intensity of an earthquake is 100 times greater than the smallest detectable wave, we can use logarithms to find its magnitude on the Richter scale: M = lo g 10 ( 100 ) = 2 . This illustrates how logarithms help us quantify and understand phenomena that vary over a wide range of values.
