# 5.08 Equations with exponentials and logarithms

Worksheet
Logarithmic equations
1

Solve for x in each of the following logarithmic equations:

a

\log_{4} 9 x = 2

b

9 \log x = 45

c

\log_{64} x = \dfrac{1}{3}

d

\log_{2} \left( 5 x - 1\right) = 6

e

\log_{7} \left( 10 x + 2311\right) = 4

f

\log_{9} \left( 5 x - 32\right) = \log_{9} 8

g

\log_{7} \left( 6 x - 6\right) = \log_{7} \left( 3 x - 2\right)

h

\log_{3} \left( 5 x - 2\right) = \log_{3} \left( 4 x + 2\right)

i

\log_{10} x + \log_{10} 9 = \log_{10} 45

j

\log \left(x + 3\right) = \log x + \log 6

k

\log_{10} 6 + \log_{10} x = \log_{10} \left(x + 8\right)

l

\log \left( 11 x - 2\right) - \log \left( 4 x - 2\right) = \log 3

m

\log \left(x + 30\right) - \log \left(x + 2\right) = \log x

n

\log_{9} \left( 4 x + 12\right) = \log_{9} \left(x + 7\right) - \log_{9} 7

2

Solve \log_{8} y = 4 for y.

Exponential equations
3

Solve for the exact value of x in each of the following exponential equations:

a

3 \left(10^{x}\right) = 6

b

\dfrac{1}{7} \left(2^{x}\right) = 3

c

3 \left(5^{ 4 x + 5}\right) + 2 = 5

d

2 \times 3^{ 5 x} + 4 = 16

4

Solve for the exact value of y in each of the following exponential equations:

a

2^{ 5 y + 2} = 3^{4 - 3 y}

b

3^{y + 5} = 6^{ 2 y}

5

Solve for x in each of the following exponential equations, correct to two decimal places:

a

5^{x} = 9

b

\left(\dfrac{1}{4}\right)^{x + 3} = \sqrt{7}

c

3^{x} = 2^{x + 1}

d

5^{x + 4} = 25^{ 8 x - 4}

6

Find the solution to 3^{x} = - 2.

7

Solve for y in each of the following exponential equations, correct to two decimal places:

a

12^{y} = 23

b

3^{y + 1} = 6

c

4^{y - 3} = 25

d

4^{y + 1} = 7

e

7^{ 4 y + 3} = 6

f

6^{y + \left( - 3 \right)} = 11^{y}

g

6^{y + 5} = 5^{y + 2}

8

Solve the following exponential equations:

a

9^{y} = 27

b
9^{x + 3} = 27^{x}
c

\left(\sqrt{6}\right)^{y} = 36

d

\left(\sqrt{2}\right)^{k} = 0.5

e

3^{ 5 x - 10} = 1

f
5^{ - 3 x -1} = 3125
g
5^{ 10 x + 33} = 125
h

\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}

i

\left(\dfrac{1}{9}\right)^{x + 5} = 81

j

\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}

k

\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}

l

8^{x + 5} = \dfrac{1}{32 \sqrt{2}}

m

30 \times 2^{x - 6} = 15

n

2^{x} \times 2^{x + 3} = 32

o
\left(2^{2}\right)^{x + 7} = 2^{3}
p
\left(2^{4}\right)^{ 2 x - 10} = 2^{2}
q

81^{x - 1} = 9^{ 3 x + 5}

r

25^{x + 1} = 125^{ 3 x - 4}

s
a^{x-1} = a^4
t

a^{x + 1} = a^{3} \sqrt{a}

u

3^{x^{2} - 3 x} = 81

v

27 \left(2^{x}\right) = 6^{x}

Inverse of exponential and logarithmic functions
9

Sketch the graph of the inverse of the following functions:

a
b
c
d
Applications
10

Find the x-coordinate of the point of intersection of the graphs of y = 2^{ 5 x} and y = 4^{x - 3}.

11

Find the value of h, given the point \left(h,\dfrac{1}{9}\right) lies on the curve y = 3^{ - x }.

12

Given the points \left(3, n\right), \left(k, 16\right) and \left(m, \dfrac{1}{4}\right) all lie on the curve with equation y = 2^{x}, find the value of:

a

n

b

k

c

m

13

A certain type of cell splits in two every hour and each cell produced also splits in two each hour. The total number of cells after t hours is given by:

N(t)=2^t

Find the time when the number of cells will reach the following amounts:

a
32
b

1024

c

4096

14

The frequency f \left(\text{Hz}\right) of the nth key of an 88-key piano is given by f \left( n \right) = 440 \left(2^{\frac{1}{12}}\right)^{n - 49}.

a

Find the frequency of the forty-ninth key.

b

Find the frequency of the 40th key to the nearest whole number.

c

Find the value of n that corresponds to the key with a frequency of 1760 \text{ Hz}.