EVALUATE LOGARITHMS WITH FRACTIONS WORKSHEET

Evaluate each of the following logarithms without the use of a calculator.

Solution

Problem 10 :

Expand the logarithmic expression :

ln (5/12x)

Solution

Problem 11 :

Use the properties of logarithms to condense each expression :

a) 3 log 4 − 2 log 𝑘

b) −5 log₂ (𝑥 +1) + 3 log₂ (6𝑥)

c) (1/3) log₄ 10 + (1/3) log₄ ℎ − 6 log₄ 𝑔

d) ln (3𝑚 +5) − 4 ln 𝑚 − ln (𝑚 − 1)

e) log 20 + 2 log (1/2) − log 𝑥 + 3 log 𝑦

Solution

Problem 12 :

Use properties of logarithms to expand each expression. The expanded logarithm expressions should have arguments with no exponent, product, or quotient

a) ln (4/5) 

b) log₆3x

c) log 7b/√c

d) log₂(m³/8n)

e) log₃[(u - 1)/(v⁵w³)]

Solution

Problem 13 :

Solve each equation.

log₄ x = log₄ 3 + log₄ (x − 2)

Solution

Problem 14 :

Solve the equation.

ln(𝑥 − 3) + ln(2𝑥 + 3) = ln (−4𝑥²)

Solution

Problem 15 :

Solve the equation.

log (3𝑥 + 2) = 1 + 𝑙𝑜𝑔 2𝑥

Solution

Solution

Problem 13 :

Solve ln (4x − 7) = ln (x + 5)

Solution

Problem 14 :

log₂(5x − 17) = 3

Solution

Problem 15 :

log₁₀(9x + 1) = 3

Solution

Problem 16 :

log₈√(1 - x) = 1/3

Solution

Problem 17 :

Solve the simultaneous equations:

log₃ x + log₅ y = 6

log₃ x - log₅ y = 2

Solution

Problem 18 :

Solve the equation

3(1 + log x) = 6 + log x

Solution

Problem 19 :

Solve log₃ [(2 + x) / (2 - x)] = 3

Solution

Without using a calculator, simplify.

Solution

Problem 9 :

Under certain conditions, the wind speed s (in knots) at an altitude of h meters above a grassy plain can be modeled by the function

s(h) = 2 ln 100h

a. By what amount does the wind speed increase when the altitude doubles?

b. Show that the given function can be written in terms of common logarithms as

s(h) = (2 /log e)  (log h + 2)

Problem 10 :

 -3 ln x = -24

Problem 11 :

4 - 3log (5x) = 16

Problem 12 :

log₃(x - 1) = -2

Problem 13 :

log₂(x² - 4) = log₂ 21

Problem 14 :

The function

s(d) = 0.159 + 0.118 log d

relates the slope, s, of a beach to the average diameter, d, in millimeters, of the sand particles on the beach. Which beach has a steeper slope: beach A, which has d = 0.0625, or beach B, which has very coarse sand with d = 1 ? Justify your decision. 

Problem 15 :

The function

S(d) = 93 log d + 65

relates the speed of the wind, S, in miles per hour, near the center of a tornado to the distance that the tornado travels, d, in miles.

a) If a tornado travels a distance of about 50 miles, estimate its wind speed near its center.

b) If a tornado has sustained winds of approximately 250 mph, estimate the distance it can travel.

Find the value of y.

Solution

Problem 13 :

Describe the similarities and difference between in solving the equations 

4^{5x - 2} = 16 and log₄(10x + 6) = 1

Then solve the each equation

Solution

Problem 14 :

For a sound with intensity I (in watts per square meter) the loudness L(I) of the sound (in decibels) is given by the function

L(I) = 10 log (I/I₀)

Where I0 is the intensity of barely audible sound (about 10^-12 watts per square meter) An artist in a recording studio turns up the volume of a track so that the intensity of the sound doubles. By how many decibels does the loudness increase ?

Solution

Problem 15 :

The length ℓ (in centimeters) of a scalloped hammerhead shark can be modeled by the function

ℓ = 266 − 219e^−0.05t

where t is the age (in years) of the shark. How old is a shark that is 175 centimeters long?

Solution