Exponential and Logarithmic Equations

Exponential and Logarithmic Equations

by: Ana Rizchel C. De Ocampo

Open Ended

What can be the other examples of EXPONENTIAL GROWTH?

Exponential Equation

It is one in which a variable occurs in the exponent.

Examples:

2^x = 128​

2^{x - 2} = 64

5^{2x - 1} = 625

​3^2x = 243

​

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

3^x = 48

16^-x = 1/64​

Solving Exponential Equations

​

1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

5^x = 625​

5^x = 5⁴

x = 4​

Multiple Choice

Solve the given exponential equation: 2^x = 64

• x = 2

• x = 4

• x = 6

• x = 8

Solving Exponential Equations

​

1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

3^2x = 81​

3^2x = 3⁴

2x = 4​

x = 2​

Multiple Choice

Solve the given exponential equation: 3^{2x - 3} = 27

• x = 3

• x = 6

• x = 9

• x = 27

Solving Exponential Equations

​

1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

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

2x + 1 = 1 - x

2x + x = 1 - 1​

3x = 0

x = 0​

Multiple Choice

Solve the given exponential equation: 8^{x - 8} = 8

• x = 1

• x = 8

• x = 0

• x = 9

Solving Exponential Equations

​

1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

(⅓)^x = 81

3^-x = 3⁴

-x = 4

x = -4​

Multiple Choice

Solve the given exponential equation: $$2^x=\frac{1}{8}$$  

• x = 3

• x = -3

• x = 4

• x = -4

Solving Exponential Equations

​

1.Rewrite both sides of the equation with the same base.

2.Set the exponents equal to one another.

3.Solve for x.​

9^{2x - 1} = 3^8x

(3²)^{2x - 1} = 3^8x

3^{4x - 2} = 3^8x

4x - 2 = 8x

-2 = 8x + 4x

-2 = 12x​​

x = ​-½

Logarithmic Equation

It is an equation that involves an expression such as log_bx. We can read it as "logarithm of x with base b" or "log base b of x."

Examples:

log₄16 = x

log​₂4 = 2

log₁₀1000 = 3​

log₃(x + 4) = 6​​

Exponential to Logarithmic

​5⁰ = 1

log₅1 = 0​

​

2⁵ = 32​

log₂32 = 5​

​3^-2 = 1/9

log₃1/9 = -2​

​

4^x = 16​

log₄16 = x​

Multiple Choice

Express the given exponential equation in its logarithmic equation:

$$6^3=216$$  

• $$\log_36=216$$  

• $$\log_3216=6$$  

• $$\log_63=216$$  

• $$\log_6216=3$$  

Multiple Choice

Express the given exponential equation in its logarithmic equation:

$$49=7^2$$  

• $$\log_749=2$$  

• $$\log_72=49$$  

• $$\log_27=49$$  

• $$\log_249=7$$  

Multiple Choice

Express the given exponential equation in its logarithmic equation:

$$m^k=5$$  

• $$\log_mk=5$$  

• $$\log_m5=k$$  

• $$\log_5k=m$$  

• $$\log_5m=k$$  

Multiple Choice

Express the given exponential equation in its logarithmic equation:

$$3^{\frac{1}{4}}=n$$  

• $$\log_3n=\frac{1}{4}$$  

• $$\log_3\frac{1}{4}=n$$  

• $$\log_n3=\frac{1}{4}$$  

• $$\log_n\frac{1}{4}=3$$  

Multiple Choice

Express the given exponential equation in its logarithmic equation:

$$3^{-4}=\frac{1}{81}$$  

• $$\log_{-4}3=\frac{1}{81}$$  

• $$\log_{-4}\frac{1}{81}=3$$  

• $$\log_3\frac{1}{81}=-4$$  

• $$\log_3-4=\frac{1}{81}$$  

Logarithmic to Exponential

​log₄64 = 3

4³ = 64​

​

log₁₄₃20,449 = 2

143² = 20,449​

​

​log_bm = n

bⁿ = m

​

log₂⅛ = -3

2^-3​ = ⅛

​

Multiple Choice

Express the given logarithmic equation in its exponential equation:

$$\log_77=1$$  

• $$1^7=7$$  

• $$7^1=7$$  

• $$7^7=1$$  

Multiple Choice

Express the given logarithmic equation in its exponential equation:

$$\log_{25}5=\frac{1}{2}$$  

• $$25^5=\frac{1}{2}$$  

• $$25^{\frac{1}{2}}=5$$  

• $$5^{\frac{1}{2}}=25$$  

• $$5^{25}=\frac{1}{2}$$  

Multiple Choice

Express the given logarithmic equation in its exponential equation:

$$\log_2\frac{1}{64}=-6$$  

• $$2^{-6}=\frac{1}{64}$$  

• $$-6^2=\frac{1}{64}$$  

• $$-6=2^{\frac{1}{64}}$$  

• $$2=-6^{\frac{1}{64}}$$  

Multiple Choice

Express the given logarithmic equation in its exponential equation:

$$\log_xy=\frac{1}{w}$$  

• $$y^{\frac{1}{w}}=x$$  

• $$y^x=\frac{1}{w}$$  

• $$x^{\frac{1}{w}}=y$$  

• $$x^y=\frac{1}{w}$$  

Multiple Choice

Express the given logarithmic equation in its exponential equation:

$$\log_2\sqrt[3]{2}=\frac{1}{3}$$  

• $$\sqrt[3]{2}=2^{\frac{1}{3}}$$  

• $$\frac{1}{3}^2=\sqrt[3]{2}$$  

• $$2^{\sqrt[3]{2}}=\frac{1}{3}$$  

• $$2^{\frac{1}{3}}=\sqrt[3]{2}$$  

Evaluating Logarithms

​Examples:

log₃27

x = ​log₃27

3^x​ = 27

3^x​ = 3³

x = 3

log₃​27 = 3

log₅625

x = log₅625

5^x = 625

5^x = 5⁴

x = 4

log₅625 = 4​

Evaluating Logarithms

​Examples:

log₅5√5

x = ​log₅5√5

5^x​ = 5√5

5^x​ = 5 ⋅ 5^½

5^x​ = 5^3/2

x = 3/2

log₁₀0.01

x = log₁₀0.01

10^x = 0.01

10^x = 10^-2

x = -2

log₁₀0.01 = -2​

​

log₅5√5 = 3/2