iGCSE (2021 Edition)

# 6.03 Solving logarithmic equations

Worksheet
Logarithmic equations
1

Solve for y in each of the following logarithmic equations:

a

\log_{10} y = 2

b

\log_{7} y = 5

c

\log_{\sqrt{49}} \left(y\right) = 3

d

\log_{y} 6 = \dfrac{1}{2}

e

\log_{y} \left(\sqrt{6}\right) = 3

f

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

2

Solve for x in each of the following logarithmic equations:

a

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

b

\log_{25} x = \dfrac{3}{2}

c

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

d

2 \log x = 4

e

\dfrac{1}{2} \log_{7} x = \dfrac{5}{8}

f

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

g

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

h

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

i

\log_{3} \left(6 - x\right) = \dfrac{1}{2}

j

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

k

\log_{7} \left(x^{3} - 15\right) = 2

l

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

m

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

n

4 \log_{3} \left( 2 x + 1\right) - 2 = 6

3

Solve for x in each of the following logarithmic equations:

a

3 \log x = \log 125

b

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

c

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

d

\log_{10} x - \log_{10} 38 = \log_{10} 37

e

\log \left(x + 7\right) = \log x + \log 3

f

\log_{10} 12 + \log_{10} x = \log_{10} \left(x + 6\right)

g

\log \left(x + 9\right) - \log 2 = \log \left( 7 x + 3\right)

h

\log \left( 14 x - 2\right) - \log \left( 5 x - 2\right) = \log 3

i

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

j

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

k

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

l

\log_{3} \left(8 x\right) - \log_{3} 40 = 2

m

\log_{4} 45 - \log_{4} \left( 9 x\right) = 2

n

\log_{5} \left(x + 12\right) - \log_{5} \left(x - 10\right) = 2

o

\log_{3} x + \log_{3} \left( 25 x\right) = 8

4

Solve for x in each of the following logarithmic equations:

a

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

b

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

c

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

d

\log_{10} x + \log_{10} \left(x - 15\right) = 2

e

\log_{4} x + \log_{4} \left(x - 60\right) = 4

f

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

g

\log_{7} \left(x + 2\right) + \log_{7} \left(x + 6\right) = \log_{7} 32