Solve $\log (7 x+3)=2$. Round to the nearest thousandth if necessary.
Answer (2)
Rewrite the logarithmic equation in exponential form: 1 0 2 = 7 x + 3 .
Simplify the equation: 100 = 7 x + 3 .
Isolate the term with x : 7 x = 97 .
Solve for x and approximate to the nearest thousandth: x = 13.857 .
Explanation
Understanding the Problem Let's solve the logarithmic equation step-by-step. First, we need to understand what the equation means. The equation lo g ( 7 x + 3 ) = 2 is a logarithmic equation where the base of the logarithm is 10. Our goal is to find the value of x that satisfies this equation.
Converting to Exponential Form To solve for x , we need to convert the logarithmic equation into its equivalent exponential form. Recall that lo g b ( a ) = c is equivalent to b c = a . In our case, the base b is 10, a is 7 x + 3 , and c is 2. Therefore, we can rewrite the equation as: 1 0 2 = 7 x + 3
Simplifying the Equation Now, let's simplify the equation. We know that 1 0 2 = 100 , so we have: 100 = 7 x + 3
Isolating the x Term Next, we want to isolate the term with x . To do this, we subtract 3 from both sides of the equation: 100 − 3 = 7 x 97 = 7 x
Solving for x Now, we solve for x by dividing both sides of the equation by 7: x = 7 97
Approximating the Value of x Finally, we need to approximate the value of x to the nearest thousandth. Dividing 97 by 7 gives us approximately 13.857. Therefore, x ≈ 13.857
Examples
Logarithmic equations are used in various real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, measuring the intensity of sound (decibels), and determining the pH of a solution in chemistry. For example, if we know the intensity of an earthquake is 100 times greater than the reference intensity, we can use logarithms to find its magnitude on the Richter scale. Similarly, in finance, logarithmic scales are used to analyze investment growth and decay over time, providing insights into percentage changes and trends.
The solution to the logarithmic equation lo g ( 7 x + 3 ) = 2 is approximately x ≈ 13.857 after converting it to exponential form and solving for x . This involves isolating the variable and calculating the value. The steps involve simplification and division.
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