Algebra 2 (HS pathway)

Logarithms and Solving Exponential and Logarithmic Equations

Algebra 2

✓By the end of this lesson students will be able to define a logarithm and convert between exponential and logarithmic forms.
✓By the end of this lesson students will be able to evaluate logarithmic expressions, including common and natural logarithms.
✓By the end of this lesson students will be able to apply the properties of logarithms (product, quotient, power, and change of base) to simplify and expand expressions.
✓By the end of this lesson students will be able to solve exponential equations using logarithms.
✓By the end of this lesson students will be able to solve logarithmic equations, checking for extraneous solutions.

Key concepts

Definition of a Logarithm

A logarithm is the inverse operation to exponentiation. It answers the question 'To what power must we raise the base to get a certain number?' The expression log_b(x) = y means that b raised to the power of y equals x. For a logarithm to be defined, the base b must be positive and not equal to 1, and the argument x must be positive.

y = log_b(x) is equivalent to b^y = x

Common Logarithm

A common logarithm is a logarithm with base 10. When no base is explicitly written, it is assumed to be 10.

log(x) = log_10(x)

Natural Logarithm

A natural logarithm is a logarithm with base e, where e is Euler's number (an irrational constant approximately equal to 2.71828). It is denoted by 'ln'.

ln(x) = log_e(x)

Properties of Logarithms

Logarithms follow specific rules that allow us to simplify, expand, and solve logarithmic expressions and equations. These properties are derived directly from the properties of exponents.

Product Property: log_b(MN) = log_b(M) + log_b(N)\nQuotient Property: log_b(M/N) = log_b(M) - log_b(N)\nPower Property: log_b(M^p) = p * log_b(M)\nChange of Base Formula: log_b(x) = log_a(x) / log_a(b) (commonly log(x)/log(b) or ln(x)/ln(b))\nInverse Properties: b^(log_b(x)) = x and log_b(b^x) = x\nSpecial Values: log_b(1) = 0 and log_b(b) = 1

Solving Exponential Equations

To solve an exponential equation where the variable is in the exponent, isolate the exponential term, then take the logarithm of both sides of the equation. Using the power property of logarithms, the exponent can be brought down, allowing you to solve for the variable. It is often convenient to use the natural logarithm (ln) or the common logarithm (log).

Solving Logarithmic Equations

To solve a logarithmic equation, first use the properties of logarithms to combine multiple logarithmic terms into a single logarithm, if necessary. Then, convert the logarithmic equation into its equivalent exponential form. After solving for the variable, it is crucial to check for extraneous solutions by ensuring that all arguments of the original logarithmic terms are positive. Logarithms of zero or negative numbers are undefined in the real number system.

Key facts to remember

1The definition of a logarithm: log_b(x) = y is equivalent to b^y = x.
2The base b must be positive and not equal to 1. The argument x must be positive.
3Common logarithm: log(x) = log_10(x). Natural logarithm: ln(x) = log_e(x).
4Product Property: log_b(MN) = log_b(M) + log_b(N).
5Quotient Property: log_b(M/N) = log_b(M) - log_b(N).
6Power Property: log_b(M^p) = p * log_b(M).
7Change of Base Formula: log_b(x) = log(x) / log(b) or ln(x) / ln(b).
8Inverse Properties: b^(log_b(x)) = x and log_b(b^x) = x. Also, log_b(1) = 0 and log_b(b) = 1.

Worked examples

Example 1

a) Evaluate log_4(64). b) Write 5^3 = 125 in logarithmic form.

Ia) To evaluate log_4(64), let y = log_4(64).

IIConvert the logarithmic equation to its equivalent exponential form: 4^y = 64.

IIIRecognize that 64 can be written as a power of 4: 64 = 4^3.

IVSo, we have 4^y = 4^3. Therefore, y = 3.

Vb) To write 5^3 = 125 in logarithmic form, identify the base, exponent, and result.

VIThe base is b = 5, the exponent is y = 3, and the result is x = 125.

VIIUsing the definition log_b(x) = y, substitute these values: log_5(125) = 3.

Answer

a) log_4(64) = 3\nb) log_5(125) = 3

Remember that a logarithm is an exponent. log_4(64) asks '4 to what power equals 64?'

Example 2

Solve 3^(2x-1) = 40 for x. Round your answer to four decimal places.

IThe exponential term is already isolated. Take the natural logarithm (ln) of both sides of the equation: ln(3^(2x-1)) = ln(40).

IIApply the Power Property of Logarithms (log_b(M^p) = p * log_b(M)) to the left side: (2x-1)ln(3) = ln(40).

IIIDivide both sides by ln(3) to isolate the term with x: 2x-1 = ln(40) / ln(3).

IVAdd 1 to both sides: 2x = (ln(40) / ln(3)) + 1.

VDivide by 2 to solve for x: x = ( (ln(40) / ln(3)) + 1 ) / 2.

VIUse a calculator to find the numerical value: ln(40) ≈ 3.688879 and ln(3) ≈ 1.098612.

VIIx ≈ (3.688879 / 1.098612 + 1) / 2

VIIIx ≈ (3.35789 + 1) / 2

9x ≈ 4.35789 / 2

10x ≈ 2.178945

11Round to four decimal places: x ≈ 2.1789.

Answer

x ≈ 2.1789

You could also use the common logarithm (log) instead of the natural logarithm; the result will be the same.

Example 3

Solve log_2(x+3) + log_2(x-3) = 4.

IFirst, identify the domain. For log_2(x+3) to be defined, x+3 > 0, so x > -3. For log_2(x-3) to be defined, x-3 > 0, so x > 3. Combining these, the valid domain for x is x > 3.

IIApply the Product Property of Logarithms (log_b(M) + log_b(N) = log_b(MN)) to combine the terms on the left side: log_2((x+3)(x-3)) = 4.

IIISimplify the argument of the logarithm: log_2(x^2 - 9) = 4.

IVConvert the logarithmic equation to its equivalent exponential form (b^y = x): 2^4 = x^2 - 9.

VCalculate 2^4: 16 = x^2 - 9.

VIAdd 9 to both sides to isolate x^2: 25 = x^2.

VIITake the square root of both sides: x = ±5.

VIIICheck for extraneous solutions using the domain x > 3:

9For x = 5: This satisfies x > 3. Substitute into the original equation: log_2(5+3) + log_2(5-3) = log_2(8) + log_2(2) = 3 + 1 = 4. This is a valid solution.

10For x = -5: This does not satisfy x > 3. If substituted, it would lead to log_2(-2) and log_2(-8), which are undefined in the real number system. Thus, x = -5 is an extraneous solution.

Answer

x = 5

Always check your solutions in the original logarithmic equation to ensure that the arguments of the logarithms remain positive. This is a critical step in solving logarithmic equations.

Common mistakes

✗Forgetting to check for extraneous solutions when solving logarithmic equations. The argument of a logarithm must always be positive.
✗Incorrectly applying logarithm properties, such as assuming log(A+B) = log(A) + log(B) or log(A)/log(B) = log(A/B).
✗Confusing the common logarithm (base 10) with the natural logarithm (base e).
✗Making algebraic errors when isolating the exponential or logarithmic term before applying inverse operations.
✗Not understanding that log_b(x) is an exponent, leading to errors in converting between logarithmic and exponential forms.

Exam tips

★Always determine the domain of logarithmic functions at the beginning of solving an equation to help identify extraneous solutions quickly.
★Memorize the logarithm properties and practice applying them in both directions (expanding and condensing expressions).
★When solving exponential equations, choose the natural logarithm (ln) or common logarithm (log) consistently for calculations, as both will yield the correct result.
★Show all steps clearly when solving equations, especially when applying logarithm properties or converting between forms, to avoid errors and earn partial credit.

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