Simple Exponent and Logarithm Equations Help (page 3)

Introduction to Simple Exponent and Logarithm Equations

Equations with exponents and logarithms come in many forms. Sometimes more than one strategy will work to solve them. We will first solve equations of the form “log = number” and “log = log.” We will solve an equation of the form “log = number” by rewriting the equation as an exponential equation.

Examples

Cancellation Law

The logarithms cancel for equations in the form “log = log” as long as the bases are the same. For example, the solution to the equation log ₈ x = log ₈ 10 is x = 10. The cancelation law a ^{log _a x} = x makes this work.

log ₈ x = log ₈ 10

8 ^{log ₈ x} = 8 ^{log ₈} 10

x = 10 (By the cancelation law)

Examples

Simple Exponent and Logarithm Equations Practice Problems

Practice

Solve for x .

1. log ₇ (2 x + 1) = 2
2. log ₄ ( x + 6) = 2
3. log 5 x = 1
4. log ₂ (8 x − 1) = 4
5. log ₃ (4 x − 1) = log ₃ 2
6. log ₂ (3 − x ) = log ₂ 17
7. ln 15 x = In( x + 4)
8. 

Solutions

Finding Approximate Solutions - Converting Exponential Equations to Logarithmic Equations

We need to use calculators to find approximate solutions for exponential equations whose base is e or 10. We will rewrite the exponential equation as a logarithmic equation, solve for x , and then use a calculator to get an approximate solution.

Examples

Finding Approximate Solutions Practice Problems

Practice

Solve for x . Give your solutions accurate to four decimal places.

1. 10 ^{3 x} = 7
2. e ^{2 x +5} = 15
3. 5000 = 2500 e ^{4 x}
4. 32 = 8 · 10 ^{6 x −4}
5. 200 = 400 e ^{−0.06 x}

Solutions

Graphing Simple Exponent and Logarithm Equations

The logarithm function f ( x ) = log _a x is the inverse of g ( x ) = a ^x. The graph of f ( x ) is the graph of g ( x ) with the x - and y-values reversed. To sketch the graph by hand, we will rewrite the logarithm function as an exponent equation and graph the exponent equation.

Examples

Graphing Simple Exponent and Logarithm Equations Practice Problems

Practice

Sketch the graph of the logarithmic function.

1. y = log _1.5 x
2. y = log ₃ x

Solutions

Finding the Domain of a Function

As long as a is larger than 1, all graphs for f ( x ) = log _a x look pretty much the same. The larger a is, the flatter the graph is to the right of x = 1. Knowing this and knowing how to graph transformations, we have a good idea of the graphs of many logarithmic functions.

• The graph of f ( x ) = log ₂ ( x − 2) is the graph of y = log ₂ x shifted to the right 2 units.
• The graph of f ( x ) = − 5 + log ₃ x is the graph of y = log ₃ x shifted down 5 units.

• log x is the graph of y = log x flattened vertically by a factor of one-third.

The domain of f ( x ) = log _a x is all positive numbers. This means that we cannot take the log of 0 or the log of a negative number. The reason is that a is a positive number. Raising a positive number to any power is always another positive number.