Problem 3 - Basic Engineering Mathematics

Question

Solve the following equations: (a) \(\lg x=3\) (b) \(\log _{2} x=3\) (c) \(\log _{5} x=-2\)

Step-by-Step Solution

Answer

(a) 1000; (b) 8; (c) \(\frac{1}{25}\)

Step 1: Understanding Logarithms

A logarithm is the power to which a number must be raised to obtain another number. The equation \( \lg x = 3 \) means we need the power to which 10 must be raised to get \( x \). Similarly, \( \log_{a} x = b \) implies \( a^b = x \).

Step 2: Solve Part (a): Convert to Exponential Form

For \( \lg x = 3 \), we convert the logarithmic form to exponential. Here, \( \lg x \) is the logarithm base 10, so \( 10^3 = x \).

Step 3: Calculate Part (a) Solution

Calculate \( 10^3 \):\[ x = 10^3 = 1000 \]

Step 4: Solve Part (b): Convert to Exponential Form

For \( \log_2 x = 3 \), convert to exponential: \( 2^3 = x \).

Step 5: Calculate Part (b) Solution

Calculate \( 2^3 \):\[ x = 2^3 = 8 \]

Step 6: Solve Part (c): Convert to Exponential Form

For \( \log_5 x = -2 \), convert to exponential: \( 5^{-2} = x \).

Step 7: Calculate Part (c) Solution

Calculate \( 5^{-2} \):\[ x = 5^{-2} = \frac{1}{25} \]

Key Concepts

Exponential Form

Exponential form is a way of expressing numbers that are too large or too small to be conveniently written or understood in decimal form. Often used in scientific notation, exponential form uses a base and an exponent.
The concept is straightforward:

• "Base" is the number that is multiplied by itself.
• "Exponent" indicates how many times the base is multiplied by itself.

For example, in the expression \(10^3\), "10" is the base and "3" is the exponent, meaning 10 is multiplied by itself three times:
\[10^3 = 10 \times 10 \times 10 = 1000\].
Logarithmic equations can be converted into exponential form to make them easier to solve. When you encounter a logarithm like \(\log_{a} b = c\), it translates into the equation \(a^c = b\). This method echoes Newton's and Leibniz's work on developing calculus, simplifying complex ideas into understandable forms.

Calculations with Exponents

When dealing with exponents, there are several rules and patterns that help simplify calculations and solve equations:

• Product of Powers: When multiplying numbers with the same base, you can add their exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Zero Exponent: Any non-zero base raised to the power of zero equals one: \(a^0 = 1\).
• Negative Exponent: Represents a reciprocal: \(a^{-n} = \frac{1}{a^n}\).

These rules are essential when the exercise requires calculating expressions such as \(10^3\), \(2^3\), and \(5^{-2}\). These calculations simplify to \(1000\), \(8\), and \(\frac{1}{25}\) respectively.

Solving Logarithmic Equations

To solve a logarithmic equation, you need to convert it into a form that is easier to handle. This transformation often involves turning the logarithm into its equivalent exponential form. Let's break down some steps that can help:

• Rewrite the Equation: Start by rewriting the logarithmic equation in its exponential form, as with \(\lg x=3\), which becomes \(10^3=x\).
• Calculate the Exponent: Once in exponential form, calculate the power to find the solution: for instance, \(10^3=1000\).
• Check for Negative Bases or Exponents: When the logarithm results in a negative exponent, as in \(\log_5 x = -2\), remember it signifies taking the reciprocal of the base (\(5^{-2} = \frac{1}{25}\)).

Logarithmic equations often serve to discover unknown values, and translating them accurately into exponential form is a powerful tool to simplify and solve them with ease. Don't forget to verify your solution by plugging it back into the original logarithmic equation to ensure it holds true.