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6.03 Solving equations with logarithms

Worksheet
Logarithmic equations
1

Solve for y in each of the following logarithmic equations:

a

\log_{7} y = 5

b

\log_{16} y = \dfrac{1}{2}

c

\log_{y} 8 = 3

d

2\log_{2} y + 3\log_{2} y = 15

e

\log_{6} y^2 = 4

f

\log_{2} y^{4} + \log_{2} y = 10

2

Solve for x in each of the following logarithmic equations:

a

\log_{2} 16 = x

b

\log_{12} 144 = x

c

\log_{3} x = 4

d

2 \log x = 4

e

\log_{0.5} x = 10

f

\log_{16} x = \dfrac{1}{4}

g

\log_{x} 64 = 2

h

\log_{x} 125 = 3

i

\log_{9} \left( 7 x + 5\right) = 4

j

\log_{6} \left( 3 x - 9\right) = 2

k

\log_{10} \left( 3 x + 982\right) = 3

l

6 \log_{4} \left( 2 x\right) - 18 = 0

m

11 \log_{5} \left(x - 12\right) = 33

n

\log_{2} \left(\sqrt{ 2 x^{3}}\right) + 1 = 4.5

o

\log_{2} \left(\log_{2} x\right) = 0

p

\log_{2} \left( 3 x + 35\right) = \log_{2} 5

q

\log_{9} \left( 10 x - 3\right) = \log_{9} \left( 5 x + 4\right)

3

Solve for x in each of the following logarithmic equations:

a

\log_{7} \left( 9 x - 2\right) = \log_{7} \left( 8 x + 9\right)

b

\log_{6} x = \sqrt{\log_{6} x}

c

\log_{2} x - 4 = \sqrt{\log_{2} x - 4}

d

3 \log x = \log 125

e

\log_{10} x + \log_{10} 6 = \log_{10} 48

f

\log_{7} 2 x + \log_{7} 3 = 3

g

\log_{4} 45 - \log_{4} 9 x = 2

h

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

i

\log x + \log \left(x + 3\right) = 1

j

\log_{3} x + \log_{3} 25 x = 8

k

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

l

\log \left(x + 12\right) - \log \left(x + 5\right) = \log x

m

\log_{8} \left( 5 x + 12\right) = \log_{8} \left(x + 6\right) - \log_{8} 3

n

\log \left(x + 5\right) + \log \left(x - 2\right) = \log 8

o

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

Exponential equations and logarithms
4

Find the interval in which the solution of the following equations will lie:

a

5^{x} = 12

b

3^{x} = 29

c

2^{x} = \dfrac{1}{13}

d
2^{x} = - 5
5

For each of the following equations:

i

Make x the subject of the equation.

ii

Evaluate x to three decimal places.

a

3^{x} = 11

b

11^{x} = 3

c

7^{x} = \dfrac{1}{13}

d

343^{x} = 12\,167

e

7^{ - x } = 21

f

4^{x - 2} = 10

g

4^{ 2 x - 8} = 70

6

For each of the following incorrect sets of working:

i

Which step was incorrect? Explain your answer.

ii

Rearrange the original equation into the form a = \dfrac{\log A}{\log B}.

iii

Evaluate a to three decimal places.

a

\begin{aligned} 9 ^ {a} &= 40 \\ \log9^{a} &= \log40 & (1)\\ a + \log 9 &= \log40 & (2)\\ a &= \log40 - \log9 & (3)\\ &\approx 0.648 & (4) \end{aligned}

b

\begin{aligned} 2 ^ {a} &= 89 \\ \log2^{a} &= \log89 & (1) \\ a \log 2 &= \log89 & (2)\\ a &= \log_{89} 2 & (3)\\ &\approx 0.154 & (4) \end{aligned}

7

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

a

8^{y} = 15

b

2^{y + 1} = 11

c

3^{y + \left( - 5 \right)} = 11^{y}

d

5^{y + \left( - 4 \right)} = 6^{y + 3}

8

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

a

2 \left(11^{x}\right) = 4

b

\dfrac{1}{3} \left(11^{x}\right) = 2

c

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

d

5^{ 5 x} = 3^{ 2 x + 3}

e

5^{x} = 2^{ 4 x + 6}

f

2^{ 3 x} = 3^{ 7 x - 1}

g

2^{ 3 x + 4} = 5^{x - 1}

h

2^{ 2 x - 7} = 7^{ 3 x + 5}

i

5 \left(2^{x}\right) = 7 - 2^{x + 3}

j

\left(7^{x}\right)^{2} - 4 \left(7^{x}\right) = 0

k

\left(5^{x}\right)^{2} + 2 \left(5^{x}\right) - 24 = 0

l

\left(3^{x}\right)^{2} - 3 \left(3^{x}\right) = 10

9

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

a

2^{6 - x} = \dfrac{1}{14}

b

3^{x} + 3 = 84

c

\left(\dfrac{1}{6}\right)^{x + 3} = \sqrt{7}

d

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

e

3 \left(5^{ 2 x - 1}\right) + 1 = 5

f

2 \log_{10}^{2} x - \log_{10} x = 6

g

\log_{10} x = x^{2} - 12 x + 8

h

2^{x} = 3 - 7^{x + 1}

i

2^{x} = \log_{10} \left(x + 5\right)

j

x = 6^{x}

k

\left(\log_{5} x\right)^{2} - \log_{5} \left(x^{6}\right) + 9 = 0

l

3 x + 2 = 4^{x}

10

The equation for the population at time t is given by Q = 30 \times 8^{ 5 t}. Make t the subject of the equation.

11

If A = 5 \log y - 50, make y the subject of the equation.

12

Rearrange the following equations to make M the subject:

a

u = 5 \log \left(\dfrac{M}{N}\right)

b

m = 2.1 \log \left(\dfrac{M}{N}\right) + 8

13

Given that \log p = \log q + r \log x, form an equation in which p is the subject, and no logarithms are involved.

14

If A = P \left(\dfrac{1 - \left(1 + r\right)^{ - t }}{r}\right), make t the subject of the equation.

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Outcomes

ACMMM153

recognise the inverse relationship between logarithms and exponentials: y=a^x is equivalent to x=log_a ⁡y

ACMMM155

solve equations involving indices using logarithms

ACMMM157

solve simple equations involving logarithmic functions algebraically and graphically

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