Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. 2(1.05)^x + 3 = 10

Verified step by step guidance

1. Start by isolating the exponential expression. Subtract 3 from both sides of the equation to get: \(2(1.05)^x = 10 - 3\).
2. Simplify the right side: \(2(1.05)^x = 7\).
3. Divide both sides by 2 to isolate the exponential term: \((1.05)^x = \frac{7}{2}\).
4. To solve for \(x\), take the natural logarithm (ln) of both sides: \(\ln\left((1.05)^x\right) = \ln\left(\frac{7}{2}\right)\).
5. Use the logarithm power rule to bring down the exponent: \(x \cdot \ln(1.05) = \ln\left(\frac{7}{2}\right)\). Then solve for \(x\) by dividing both sides by \(\ln(1.05)\): \(x = \frac{\ln\left(\frac{7}{2}\right)}{\ln(1.05)}\).

Key Concepts

Exponential Equations

An exponential equation is one in which the variable appears in the exponent, such as 2(1.05)^x + 3 = 10. Solving these requires isolating the exponential expression before applying logarithms or other methods to find the variable.

Isolating the Exponential Term

To solve an exponential equation, first isolate the term containing the exponent by performing inverse operations like subtraction and division. For example, subtract 3 and then divide by 2 to get (1.05)^x alone.

Using Logarithms to Solve for the Exponent

Once the exponential term is isolated, apply logarithms (common or natural) to both sides to bring the exponent down. This allows solving for x by using the property log(a^x) = x log(a), enabling calculation of the variable.