Laws of Logarithms

Solving Logarithmic Equations

By Dr KEE CH

Learning objectives:

1. Students will be able to solve logarithmic equations using the laws of logarithms.

2. Students will be able to demonstrate the seven laws of logarithms confidently.

3. Students will be able to conceptualize logarithmic functions as inverse of exponential functions.

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Law 1: Product Rule

log_b​(M⋅N)=log_bM+log_bN

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Expand $$\log_3\left(27x^2y^5\right)$$  

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Expand $$\log_212w^5\sqrt[]{y}$$  

Law 2: Quotient Rule

log_b​(M/N)=log_bM-log_bN

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Evaluate $$\log_3162-\log_32$$  

Law 3: Power Rule

log_b​(M^k)= k⋅log_bM

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Evaluate $$\log_28$$  

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Evaluate $$\log_28+\log_24$$  

Wait! There's more...

Not really a "law" but useful to know

log_b​(1)= 0

Not really a "law" but useful to know

log_b​(b)= 1

Not really a "law" but useful to know

log_b​(b^k)= k

Not really a "law" but useful to know

b^logb​(k)= k

Change of Base Rule

log_ba=log_x b / log_x a

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Evaluate $$\log_58$$  by changing the base to $$\log_{10}$$  

How to solve logarithmic equation?

log₃x = log₃7 + log₃3​

If log₁₀8 = x and log₁₀3 = y

Express log₁₀24 in terms of x and y only​

If log₁₀8 = x and log₁₀3 = y

Express log₁₀720 in terms of x and y only​

​You try!

Check your understanding

Multiple Choice

If 3^x = y, which of the following is true?

• log₃y = x
• log₃x = y
• log_xy = 3
• log_y3 = x

Multiple Choice

If log_a8 = b, which of the following is true?

• a^b = 8
• b^a = 8
• a⁸ = b
• 8^b = a

Multiple Choice

Simplify log(12) + log (6)

• log(72)
• log(18)
• log(6)
• log(2)

Multiple Choice

Simplify: log(3) + 2log(5)

• log(75)
• 2log(15)
• log(30)
• log(13)

Multiple Choice

Simplify: 2log(x) + 3log(xy)

• log(x⁵y³)
• log(5xy)
• 5log(x²y)
• log(6x²y)

Multiple Choice

Simplify: log(24) - log(8)

• log(3)
• log(16)
• 4log(2)
• -log(32)

Multiple Choice

Simplify: log(16) - 3log(2)

• log(2)
• log(10)
• -3log(8)
• 3log(14)

Multiple Choice

Simplify: 3log(xy) - 2log(x)

• log(xy³)
• log(y)
• 1.5log(x²y)
• Can't be simplified

Multiple Choice

What is the most simple answer possible:
log_x(x³y) - log_x(xy)

• log_x(x²)
• 2
• log_x(x⁴-y²)
• x

Multiple Choice

Solve for x:
log(3x-1) - log(5) = 2

• x = 167
• x = 500
• x = 2
• x = 3.67

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Evaluate $$\log_5500-2\log_52+\log_432+\log_48$$  

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Expand $$\log_7\left(\frac{49m^6}{k^3}\right)$$