Problem 75 - Calculus for Biology and Medicine

Question

Which real number \(x\) satisfies (a) \(\log _{4} x=-2 ?\) (b) \(\log _{1 / 3} x=-3\) ? (c) \(\log _{10} x=-2 ?\)

Step-by-Step Solution

Answer

(a) \(x = \frac{1}{16}\); (b) \(x = 27\); (c) \(x = \frac{1}{100}\).

Step 1: Understand Logarithmic Function

Recall the definition of a logarithm: If \( \log_b a = c \), then \( b^c = a \). This means that the logarithmic equation gives us the exponent (\(c\)) to which the base (\(b\)) must be raised to get the number \(a\).

Step 2: Solve Part (a)

We have \( \log_4 x = -2 \). According to the logarithm definition, this implies \( 4^{-2} = x \). Calculate \( 4^{-2} \): \( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \). Thus, \( x = \frac{1}{16} \).

Step 3: Solve Part (b)

We have \( \log_{1/3} x = -3 \). This translates to \((1/3)^{-3} = x\) using the rule of logarithms. Calculate \((1/3)^{-3}\): \((1/3)^{-3} = 3^3 = 27\). So, \(x = 27\).

Step 4: Solve Part (c)

We have \( \log_{10} x = -2 \). From the definition of a logarithm, \(10^{-2} = x\). Calculate \(10^{-2}\): \(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\). Therefore, \(x = \frac{1}{100}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is raised to the power of an exponent. This operation is expressed in the form of \(b^c\), which means multiplying the base \(b\) by itself \(c\) number of times. Here's a quick breakdown of the key points related to exponentiation:

• The base is the number that is being multiplied.
• The exponent tells us how many times to use the base in a multiplication.
• When the exponent is positive, the operation results in repeated multiplication.
• When the exponent is zero, the result is always 1, as in \(b^0 = 1\).
• If the exponent is negative, the operation involves taking the reciprocal of the base raised to the absolute value of the exponent, such as \(b^{-c} = \frac{1}{b^c}\).

Understanding exponentiation is crucial for solving logarithmic equations, as logarithms are the inverse processes of exponentiation, where we find what power we need to raise the base of the exponent to obtain a specific number.

Solve Logarithmic Equations

Solving logarithmic equations involves reversing the logarithm operation to find the unknown quantity, which is typically done by applying exponentiation. Here are the generalized steps to solve a logarithmic equation \(\log_b x = c\):

• Step 1: Recall the definition of logarithm: if \(\log_b a = c\), then \(b^c = a\). This states that the equation \(\log_b x = c\) can be rewritten in exponential form as \(b^c = x\).
• Step 2: Convert the logarithmic equation to an exponential equation. Essentially, this means using the concept of exponentiation to solve the logarithm. For instance, in \(\log_4 x = -2\), express this as \(4^{-2} = x\).
• Step 3: Calculate the value of the base raised to the given power. For \(4^{-2}\), compute it as \(\frac{1}{4^2} = \frac{1}{16}\).

By following these steps, you can find the value of \(x\), turning a logarithmic expression into a more familiar exponential problem, which simplifies the solving process.

Real Number Solutions

In mathematics, real numbers include all the numbers on the number line, covering both positive and negative numbers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\). Finding real number solutions means ensuring that the solutions derived from equations actually exist within this set.

• When dealing with logarithmic equations, solutions must be positive real numbers. This is because the logarithm function is only defined for positive real input values.
• Real number solutions can be rational or irrational, and they must satisfy the original equation when substituted back into it.
• In exercises involving logarithms, such as \(\log_{1/3} x = -3\), it's crucial to determine if the computation, for example substituting back, yields x to be a valid real number like \(27\).

For the problems given, the solutions \(\frac{1}{16}, 27,\) and \(\frac{1}{100}\) are all valid real numbers as they fit the definition and satisfy their respective equations.