Скачать или смотреть An amazing trick for solving this exponential equation

After watching this video, you would be able to solve this exponential equation or other similar exponential equations with ease.
Exponential Equations
Exponential equations involve variables in the exponent. They can be solved using various methods, including:

Methods
1. *Inspection*: Simple equations can be solved by inspection.
2. *Logarithms*: Taking logarithms of both sides can help solve more complex equations.
3. *Substitution*: Substituting a new variable can simplify the equation.

Examples
1. 2^x = 8 implies 2^x = 2^3 implies x = 3
2. e^x = 10 implies x = ln(10)

Tips
1. *Use logarithm properties*: To simplify equations and solve for the variable.
2. *Check the solution*: Verify the solution satisfies the original equation.

Types of Exponential Equations
1. *Simple exponential equations*: a^x = b
2. *Exponential equations with different bases*: a^x = b^y

Applications
1. *Population growth*: Modeling population growth and decay.
2. *Finance*: Calculating compound interest and investments.
3. *Science*: Modeling exponential growth and decay in various fields.
Exponential Equations with Different Bases
To solve exponential equations with different bases, we can use:

Methods
1. *Logarithms*: Take the logarithm of both sides.
2. *Change of base formula*: Convert to a common base.

Example 1: Using Logarithms
Solve: 2^x = 3^(x-1)

Solution
1. Take the logarithm of both sides: log(2^x) = log(3^(x-1))
2. Use logarithm properties: x*log(2) = (x-1)*log(3)
3. Solve for x: x_log(2) = x_log(3) - log(3)
4. x*(log(2) - log(3)) = -log(3)
5. x = -log(3) / (log(2) - log(3))

Example 2: Using Change of Base Formula
Solve: 2^x = 3^x

Solution
1. Take the logarithm of both sides: log(2^x) = log(3^x)
2. Use logarithm properties: x_log(2) = x_log(3)
3. Solve for x: x*(log(2) - log(3)) = 0
4. x = 0 (since log(2) ≠ log(3))

Tips
1. *Choose a base*: For logarithms, choose a convenient base (e.g., natural logarithm or common logarithm).
2. *Simplify*: Use logarithm properties to simplify equations.

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