Question

Solve (a) \(10^{x}=7\) (b) \(10^{x}=70\) (c) \(10^{x}=17\) (d) \(10^{\mathrm{x}}=0.7000\)

Short answer

(a) \(10^{x}=7\) (b) \(10^{x}=70\) (c) \(10^{x}=17\) (d) \(10^{\mathrm{x}}=0.7000\) Answer: (a) \(x\approx 0.8451\) (b) \(x\approx 1.8451\) (c) \(x\approx 1.2304\) (d) \(x\approx -0.1549\)

Step-by-step solution

Rewrite exponential equations as logarithmic equations

To rewrite an exponential equation with base 10 as a logarithmic equation, we can use the formula: \(\log_{a}^{b} = x\), where a is the base, b is the exponent, and x is the result. (a) \(10^{x}=7\) \(\log_{10}(7)=x\) (b) \(10^{x}=70\) \(\log_{10}(70)=x\) (c) \(10^{x}=17\) \(\log_{10}(17) = x\) (d) \(10^{\mathrm{x}}=0.7000\) \(\log_{10}(0.7000) = x\)

Solve for x using logarithm properties.

To solve for x, we will use log properties, specifically the change of base formula to calculate the base 10 logarithms. Change of base formula: \(\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}\) For the base 10 logarithm, we will change the base to the natural logarithm (base e) because it is easily available on most calculators. The change of base formula will be as follows: \(\log_{10}(b) = \frac{\ln(b)}{\ln(10)}\) (a) \(x = \log_{10}(7) = \frac{\ln(7)}{\ln(10)} \approx 0.8451\) (b) \(x = \log_{10}(70) = \frac{\ln(70)}{\ln(10)} \approx 1.8451\) (c) \(x = \log_{10}(17) = \frac{\ln(17)}{\ln(10)} \approx 1.2304\) (d) \(x = \log_{10}(0.7000) = \frac{\ln(0.7000)}{\ln(10)}\approx -0.1549\) The solutions for each part are: (a) \(x\approx 0.8451\), (b) \(x\approx 1.8451\), (c) \(x\approx 1.2304\), and (d) \(x\approx -0.1549\).

Key concepts

Exponential Equations

Exponential equations involve expressions where the variable is in the exponent. Things like \(10^x = 7\) or \(10^x = 70\) are examples of exponential equations. These types of equations can be quite challenging to solve directly, especially when the exponent is unknown, which is why they often require special methods of solving, such as logarithms. Exponential equations typically have a common form:

• \(a^x = b\)

Where \(a\) is the base, \(x\) is the variable exponent, and \(b\) is a constant on the other side of the equation. When the base \(a\) is a known value like 10, it's easier to convert these equations into logarithmic form to find the value of \(x\). By converting an exponential equation into a logarithmic form, you get an equation that's easier to work with, especially if you want to use a calculator. For example, the equation \(10^x = 7\) becomes \(x = \log_{10}(7)\), which can be computed directly using a scientific calculator.

Change of Base Formula

The change of base formula is an important tool when dealing with logarithms, particularly when you need to calculate logarithms that aren't as straightforward due to the base involved. You typically encounter this when the base of the logarithm isn't compatible with your calculator's default setting (usually base 10 or base \(e\)). The formula is: \[\log_{a}(b) = \frac{\log_{c}(b)}{\log_{c}(a)}\]This formula allows you to convert any logarithmic base \(a\) to a more convenient base \(c\), which is why it's especially useful when using a calculator.Most calculators can easily handle natural logarithms (\(\ln\)) or logarithms with base 10 (\(\log_{10}\)). So, if you have an equation like \(10^x = 7\) and need to solve for \(x\), it would convert to:\[\log_{10}(7) = \frac{\ln(7)}{\ln(10)}\]This makes the calculation much simpler, since \(\ln\) is readily available on most scientific calculators.

Natural Logarithm

Natural logarithms are logarithms with a base of \(e\), where \(e\) is a mathematical constant approximately equal to 2.71828. The notation for a natural logarithm is \(\ln\), which stands for "logarithmus naturalis." Using \(\ln\) is very common due to its useful properties, particularly in calculus and mathematical analyses involving growth and decay. Natural logarithms simplify many mathematical expressions and are implemented on virtually all calculators. When converting an expression like \(\log_{10}(b)\) using the natural logarithm, the expression uses the change of base formula: \[\log_{10}(b) = \frac{\ln(b)}{\ln(10)}\]This formula signifies that we can switch to natural logarithms, ensuring we can handle calculations effectively.Natural logarithms offer a powerful and efficient way to compute logarithms, serving well in both theoretical and practical settings. Whether you're dealing with exponential growth or solving logarithmic equations, \(\ln\) often provides a straightforward path to a solution.