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Solving equations with logarithms

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recap

Estimate the answers and then use your calculator to solve:

3^x = 10

4^x = 80

10^x = 5

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revision

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3^x = 3⁶ 4^x = 16 5^x = ¹/₂₅ 100^2x = 10

⇒ x = 6 ⇒ 4^x = 4² ⇒ 5^x = 5^-2 ⇒ (10²)^2x = 10

⇒ x = 2 ⇒ x = -2 ⇒ 10^4x = 10

⇒ 4x = 1

⇒ x = ¼

How these equations have been solved?

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Write 16 as a power of 4

The powers must be equal, as we have the same base

Subtract 1

Divide by 2

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Write both sides as powers of 3

Multiply the powers on the left side

Now equate the powers

Subtract 2

Divide by 2

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Your turn.

Solve these equations by equating the powers of equal bases (no calculator).

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Watch the video

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Solve the equations

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Solve the equations

nb log x means log₁₀ x

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Watch the video

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Practice 1

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Solve

5^2x+1 = 7^x

algebraically

graphically

interpret

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Solve simultaneously

2log₂x = y

log₂x = y + 4

algebraically

graphically

interpret

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Applied

The compound interest on a savings account is 4.2% per annum. After how many years would an initial investment of £2000 exceed £8000?

A colony of bacteria has a growth factor of 4 per hour. Initially there are 150 bacteria present. How long will it take the number of bacteria to reach 150 000?

A capacitor is a device for storing electrical charge. Its charge decays exponentially, so that after 1 second it has lost half its charge. It stores 500 coulombs of charge at the start. How long will it take the charge stored to drop to one tenth of the initial value?

A fungus has an initial mass of 10kg and growth factor 1.3 per day. Harvesting occurs when the mass of the fungus is 250kg. How long will the farmer have to wait to harvest?

A sample of radium decays at a rate which is proportional to its mass. The mass of radium, m, is given by the equation m = m_oe^-kt where m_o is the initial mass, k is the decay constant and t is the time in years. If the half-life is 1600 years find the value of k and the percentage rate of decrease per century.