- Apply the definition of the common logarithm to the equation \(10^x = 3.66\) to get \(x = \log_{10}(3.66)\).
- Utilize the change-of-base formula to express the solution in terms of natural logarithms: \(x = \frac{\ln(3.66)}{\ln(10)}\).
- Calculate the decimal value to find \(x \approx 0.5635\).
- Final Answer: \(\boxed{\log(3.66) \approx 0.5635}\).
Explanation
- Isolating x using the Logarithm Definition
To solve the exponential equation \(10^x = 3.66\), we first recognize that the variable \(x\) is in the exponent. The most direct way to isolate \(x\) when the base is \(10\) is to apply the common logarithm (\(\log_{10}\)) to both sides of the equation. By the definition of the logarithm, the statement \(b^y = z\) is equivalent to \(y = \log_b(z)\). In our case, \(b = 10\), \(y = x\), and \(z = 3.66\), so we can write \(x = \log_{10}(3.66)\).
- Expressing the Solution using Natural Logarithms
Alternatively, we can express the solution using natural logarithms (\(\ln\)). Applying the property \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\), we convert the common logarithm \(\log_{10}(3.66)\) into natural logarithms:
\[x = \frac{\ln(3.66)}{\ln(10)}\]
This expression is equivalent to our previous result and is often useful in scientific calculations.
- Calculating the Numerical Value
Finally, we calculate the decimal approximation for the value of \(x\). Using numerical methods, we find that:
\[x = \log_{10}(3.66) \approx 0.563481\]
Rounding to a standard number of decimal places, we have \(x \approx 0.5635\). Thus, the solution set is \(\boxed{\log(3.66) \approx 0.5635}\).
Examples
Exponential equations appear frequently in fields like chemistry and biology, for instance, when calculating the pH of a solution or modeling bacterial population growth. In chemistry, the pH of a liquid is defined as \(pH = -\log_{10}([H^+])\), which can be rewritten as the exponential form \(10^{-pH} = [H^+]\). Solving such equations allows us to determine the concentration of hydrogen ions based on the acidity of a solution, helping scientists understand how chemical environments affect biological processes.