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Logarithmic properties and equations

MAT1510 - Precalculus Mathematics A · Exponential and Logarithmic Functions

Logarithmic Properties and Equations

Logarithmic functions are the inverses of exponential functions. Understanding their properties is essential for solving equations and inequalities involving logarithms. This section will cover the key properties of logarithms and how to apply them in solving logarithmic equations.

Logarithmic Properties

Logarithms have several important properties that simplify calculations. The three main properties are:

  • Product Property: logb(xy) = logb(x) + logb(y)
  • Quotient Property: logb(x/y) = logb(x) - logb(y)
  • Power Property: logb(xn) = n × logb(x)

Remember: These properties hold true for any positive base b, where b ≠ 1.

Example: Using Logarithmic Properties

Let us calculate log2(32) using the properties of logarithms.

  1. First, express 32 as a power of 2: 32 = 25.
  2. Using the Power Property, we have:
log2(32) = log2(25)
  1. Applying the Power Property:
log2(32) = 5 × log2(2)
  1. Since log2(2) = 1, we continue:
log2(32) = 5 × 1 = 5

Thus, log2(32) = 5.

Change of Base Formula

Sometimes, you may need to calculate logarithms with a base that is not readily available on your calculator. The Change of Base Formula allows you to convert logarithms to a different base:

logb(x) = logk(x) / logk(b), where k is any positive number (commonly 10 or e).

Example: Change of Base Formula

Calculate log3(9) using the Change of Base Formula.

  1. Using base 10, we write:
log3(9) = log10(9) / log10(3)
  1. Using a calculator, we find:
log10(9) ≈ 0.9542 and log10(3) ≈ 0.4771
  1. Now, divide the two results:
log3(9) ≈ 0.9542 / 0.4771 ≈ 2

Thus, log3(9) = 2, which is correct since 32 = 9.

Solving Logarithmic Equations

Logarithmic equations can be solved using the properties of logarithms. To solve an equation, isolate the logarithm and then rewrite it in exponential form.

Example: Solving a Logarithmic Equation

Consider the equation:

log2(x) = 5

  1. First, rewrite the equation in exponential form:
x = 25
  1. Calculate the right side:
x = 32

Thus, the solution is x = 32.

Example: A More Complex Logarithmic Equation

Now, let’s solve:

log3(x + 1) = 2

  1. Rewrite in exponential form:
x + 1 = 32
  1. Calculate the right side:
x + 1 = 9
  1. Isolate x:
x = 9 - 1
x = 8

Thus, the solution is x = 8.

Logarithmic Inequalities

Logarithmic inequalities can also be solved using the properties of logarithms. The approach is similar to solving equations, but you must consider the direction of the inequality.

Example: Solving a Logarithmic Inequality

Consider:

log2(x) > 3

  1. Rewrite in exponential form:
x > 23
  1. Calculate the right side:
x > 8

Thus, the solution is x > 8.

Common Mistakes

Watch out: When solving logarithmic equations, ensure that the solution is within the domain of the logarithm. For example, logb(x) is only defined for x > 0.

Summary

  • Logarithmic properties include product, quotient, and power properties.
  • The Change of Base Formula allows conversion between different bases.
  • Logarithmic equations can be solved by rewriting them in exponential form.
  • Logarithmic inequalities require careful consideration of the inequality direction.

Check your understanding

  1. Using the properties of logarithms, simplify log5(25) + log5(5).
  2. Use the Change of Base Formula to calculate log2(16).
  3. Solve the equation log4(x - 3) = 1.
  4. Determine the solution for the inequality log10(x) < 2.