Logarithm Properties Lesson

Logarithm Properties Lesson

Follows Topic 14 p.13-14, 21-23

What is the general form of the logarithmic function? 

y = a log_b(x - h) + k

Multiple Choice

What are the effects of changing the parameter h in the general logarithmic function?

• Horizontal shift

• Vertical Shift

• Stretch

• Compression

Multiple Choice

Does a change in h cause a change in the domain and range?

• Yes, because the shift is left/right x-values

• No, because the shift is left/right y-values

 Discuss any changes to asymptotes as well.

The asymptote shifts in the same manner, which results in a domain change. The equation of the asymptote is x = h and the domain is all real numbers greater than h.

Multiple Select

What are the effects of changing the parameter a in the general logarithmic function? Select all that apply.

• Horizontal shift

• Vertical Shift

• Stretch

• Compression

• Reflection

Multiple Choice

Does a change in a cause a change in the domain and range?

• Yes

• No

Discuss any changes to the asymptote as well. 

When a is negative, there is a reflection over the x-axis. Since the range of the parent function is all real numbers, a vertical stretch, compression, or reflection does not affect it. The domain is also not affected, so the asymptote is not affected either. 

Multiple Choice

What are the effects of changing the parameter k in the general logarithmic function?

• Horizontal shift

• Vertical Shift

• Stretch

• Compression

Multiple Choice

Does a change in k cause a change in the domain and range?

• Yes, because the shift is up/down y -values, will change the range

• No, the domain and range will remain the same

Why or why not? Discuss any changes to the asymptote as well.  

Since the range of the parent function is all real numbers, a vertical shift does not affect it. The domain and the asymptote are also not affected.

p. 21

Properties of Exponents/Logs

Open Ended

b⁰ =

Multiple Choice

bⁿ x b^m =

• b^nm

• b^{n + m}

• b^{n / m}

• b^{n - m}

Multiple Choice

bⁿ / b^m =

• b^nm

• b^{n + m}

• b^{n / m}

• b^{n - m}

Multiple Choice

(bⁿ)^m =

• b^nm

• b^{n + m}

• b^{n / m}

• b^{n - m}

Multiple Choice

1. Rewrite the statement b⁰ = 1 in logarithmic form:

• log_b0=1

• log₀b =1

• log₁0 =b

• log_b1 = 0

Summarize this logarithmic property in words.

This property says that the log of 1, in any base, is always equal to 0.

Multiple Choice

1. Rewrite the statement b¹ = b in logarithmic form.

• log_bb=1

• log₁b = b

• log_b1=b

Multiple Choice

1. Rewrite the statement bⁿ • b^m = b^{n + m} in logarithmic form. Let bⁿ = p and b^m = q.

• logb (p •q) = logb p + logb q

• logb (p + q) = logb p x logb q

• logb (p + q) = logb p + logb q

• logb (p x q) = logb p x logb q

In your own words, state the logarithmic property you discovered in the previous question.

The log of a product is equal to the sum of the logs of each factor.

Multiple Choice

Rewrite the statement bⁿ /b^m = b^{n - m} in logarithmic form. Again, let bⁿ = p and b^m = q.

• log_b (p / q) = log_b p − log_b q

• log_b (p / q) = log_b p / log_b q

• log_b (p - q) = log_b p - log_b q

• log_b (p - q) = log_b p / log_b q

n your own words, state the logarithmic property you discovered in the previous question.

The log of a quotient is equal to the difference of the logs.

Multiple Choice

Rewrite the expression log(3x) as an equivalent logarithmic expression using only addition.

• log 3 + log x

• log 3 / log x

• log 3 - log x

• (log 3)(log x)

Multiple Choice

Rewrite the expression below as an equivalent logarithmic expression using only subtraction.

log₄(10 / 4)

• log₄10 - log₄y

• log₄(10 - y)

Multiple Choice

Apply the properties of logarithms to simplify each expression for x ≥ 0.

log_x(27) − log_x(3) =

• log _x 9

• log _x 24

• log _x 30

• log _x 81

Multiple Choice

Apply the properties of logarithms to simplify each expression for x ≥ 0.

log₁₀(2) + log₁₀(5) =

• log₁₀10

• log₁₀7

• log₁₀2/5

• log₁₀ 3

Multiple Choice

Apply the properties of logarithms to simplify each expression for x ≥ 0.

log (2 / x) + log (x / 2)

• log 1

• log 4x²

• log 2x

• log 0

Multiple Choice

Rewrite the statement (bⁿ ) ^m = b^nm in logarithmic form. Again, let bⁿ = p and b^m = q.

• log_b (p) ^m = mlog_b p

• log_b (pm) = mlog_b p

• log_b (p) ^m = plog_b m

• log_b (m) ^p = mlog_b p

In your own words, state the logarithmic property you discovered in the previous question.

The log of an exponential expression can be written as the product of the exponent and the log of the base of that exponential expression.

Multiple Choice

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x, and log y.

log (x³ / y²)

• 3log x - 2 log y

• x log 3 + 2 log y

• log 3 + log x + log y + log 2

• 3 log x + 2 log y

Multiple Choice

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x, and log y.

log 7x⁴

• log7 + 4log x

• 7 log x + log 4

• log 7 + x log 4

• log 7 - 4 log x

Multiple Choice

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x, and log y.

log xy³

• log x + 3 log y

• log x + y log 3

• 3 log x + log y

• log x - 3 log y