# If 2 x 3 is 3 x

### Exponential equations

You can already solve linear equations like $$ 3x + 2 = 4 $$ or quadratic equations like $$ x ^ 2-x-2 = 0 $$.

The variable $$ x $$ can also be in the exponent:

$$ a ^ x = b $$ with $$ a, b \ in RR $$, $$ a ne 0 $$

**example**:

$$ 2 ^ x = 8 $$

You can often solve simple exponential equations like $$ 2 ^ x = 8 $$ in your head: $$ 2 $$ high what is $$ 8 $$?

$$ x = 3 $$ is the solution to the equation.

Sample:

$$2^3 =?$$

That's $$ 8 $$. Fits.

For difficult exponential equations you need the logarithm. Remember: $$ b ^ x = y $$ means the same as $$ log_b (y) = x $$.

**example**:

$$ 2 ^ x = 32 $$ is $$ log_2 (32) $$

$$ log_2 (32) = 4 $$, since $$ 2 ^ 4 = 32 $$

Let $$ y $$ and $$ b ≠ 1 $$ two positive numbers.

Equations in which the variable $$ x $$ is in the exponent are called exponential equations.

### Solve exponential equations with the logarithm

Here's what to do when you can't solve the exponential equation in your head. Log the equation on both sides. You can choose any base of the logarithm. Then apply the logarithmic laws.

**Example:**

$$ 3 ^ x = 2187 $$

$$ log (3 ^ x) = log (2187) $$

$$ x * log (3) = log (2187) $$

$$ x = log (2187) / log (3) $$

You can now type that into the calculator. It comes out: $$ x = 7 $$

Sample:

$$3^7=?$$

That's $$ 2187 $$. Correctly calculated!

Logarithmic Laws:

For logarithms based on $$ b $$ with $$ b ≠ 1 $$ and $$ b> 0 $$ and for positive real numbers $$ u $$ and $$ v $$ as well as a real number $$ r $$ applies:

1. $$ log_b (u * v) = log_b (u) + log_b (v) $$

2. $$ log_b (u / v) = log_b (u) -log_b (v) $$

3. $$ log_b (u ^ r) = r * log_b (u) $$

### Sometimes the equations still need to be changed ...

Exponential equations can have a factor.

Like equations you already know, you bring exponential equations to the form $$ a ^ x = b $$.

$$ c * a ^ x = b $$

Put the equation in the form $$ a ^ x = b $$. So divide by $$ c $$.

**Example:**

$$ 2 * 2 ^ x = 16 $$ | $$: 2 $$

$$ 2 ^ x = 8 $$ | $$ log $$

$$ log (2 ^ x) = log (8) $$ | $$ 3. $$ Law of logarithms

$$ x * log (2) = log (8) $$ | $$: log (2) $$

$$ x = log (8) / log (2) = 3 $$

Sample:

$$2^3=?$$

That is $$ 2 * 8 = 16 $$. Correctly calculated!

Exponential equations can have additional factors or summands. Then always first bring the equation to the form $$ a ^ x = b $$.

Logarithmic Laws:

For logarithms based on $$ b $$ with $$ b ≠ 1 $$ and $$ b> 0 $$ and for positive real numbers $$ u $$ and $$ v $$ as well as a real number $$ r $$ applies:

1. $$ log_b (u * v) = log_b (u) + log_b (v) $$

2. $$ log_b (u / v) = log_b (u) -log_b (v) $$

3. $$ log_b (u ^ r) = r * log_b (u) $$

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### $$ x $$ on both sides of the exponential equation

### One factor

$$ c * a ^ x = b ^ x $$

Divide the equation by $$ a ^ x $$ and apply the 4th power law.

**Example:**

$$ 8 * 8 ^ x = 16 ^ x $$ $$ |: 8 ^ x $$

$$ 8 = (16 ^ x) / (8 ^ x) $$ $$ | 4. $$ power law

$$ 8 = (16/8) ^ x $$

$$ 8 = 2 ^ x $$ $$ | log $$

$$ log (8) = log (2 ^ x) $$ $$ | 3. $$ Law of logarithms

$$ log (8) = x * log (2) $$ $$ |: log (2) $$

$$ x = log (8) / log (2) = 3 $$

Sample:

$$8*8^3=4096=16^3$$

Pooh, calculated correctly!

### Two factors

$$ c * a ^ x = d * b ^ x $$

Divide the equation by $$ a ^ x $$ and by $$ d $$ and then apply the 4th power law.

**Example:**

$$ 32 * 8 ^ x = 4 * 16 ^ x $$ $$ |: 8 ^ x |: 4 $$

$$ 8 = (16 ^ x) / (8 ^ x) $$ $$ | 1. $$ power law

$$ 8 = (16/8) ^ x $$

$$ 8 = 2 ^ x $$ $$ | log $$

$$ log (8) = log (2 ^ x) $$ $$ | 3. $$ Law of logarithms

$$ log (8) = x * log (2) $$ $$ |: log (2) $$

$$ x = log (8) / log (2) = 3 $$

Sample:

$$32*8^3=4*16^3???$$

$$16384=16384$$

Great, calculated correctly!

Logarithmic Laws:

For logarithms based on $$ b $$ with $$ b ≠ 1 $$ and $$ b> 0 $$ and for positive real numbers $$ u $$ and $$ v $$ as well as a real number $$ r $$ applies:

1. $$ log_b (u * v) = log_b (u) + log_b (v) $$

2. $$ log_b (u / v) = log_b (u) -log_b (v) $$

3. $$ log_b (u ^ r) = r * log_b (u) $$

Power laws:

For powers with the bases $$ a $$ and $$ b $$ and for rational numbers $$ x, y $$:

1. $$ (a ^ x) / (b ^ x) = (a / b) ^ x $$

2. $$ (a ^ x) ^ y = a ^ (x * y) $$

### Even more going on in the exponent

### Sum in the exponent

$$ a ^ (x + e) = b $$

Apply the 1st power law and then calculate as usual.

**Example:**

$$ 6 ^ (x + 2) = 360 $$ $$ | 3rd $$ power law

$$ 6 ^ x * 6 ^ 2 = 360 $$ $$ |: 6 ^ 2 $$

$$ 6 ^ x = 360 / (6 ^ 2) $$

$$ 6 ^ x = 10 $$ $$ | log $$ $$ | 3. $$ Law of logarithms

$$ x * log (6) = log (10) $$ $$ |: log (6) $$

$$ x = log (10) / log (6) approx1.285 $$

Sample:

$$6^(1,285+2)=???$$

That's about $ 360 $. Correctly calculated!

### Product in exponent

$$ a ^ (e * x) = d * b ^ x $$

Apply the 2nd power law and then calculate as usual.

**Example:**

$$ 3 ^ (2 * x) = 4 * 5 ^ x $$ $$ | 2. $$ power law

$$ (3 ^ (2)) ^ x = 4 * 5 ^ x $$ $$ |: 5 ^ x $$

$$ (9 ^ x) / (5 ^ x) = 4 $$

$$ 1,8 ^ x = 4 $$ $$ | log $$ $$ | 3. $$ Law of logarithms

$$ x * log (1.8) = log (4) $$ $$ |: log (1.8) $$

$$ x = log (4) / log (1.8) approx 2.358 $$

Sample:

$$3^(2*2,358)=4*5^2,358???$$

That's true, if you round up the results. Correctly calculated!

For logarithms based on $$ b $$ with $$ b ≠ 1 $$ and $$ b> 0 $$ and for positive real numbers $$ u $$ and $$ v $$ as well as a real number $$ r $$ applies:

1. $$ log_b (u * v) = log_b (u) + log_b (v) $$

2. $$ log_b (u / v) = log_b (u) -log_b (v) $$

3. $$ log_b (u ^ r) = r * log_b (u) $$

Power laws:

For powers with the bases $$ a $$ and $$ b $$ with and for rational numbers $$ x, y $$:

1. $$ (a ^ x) / (b ^ x) = (a / b) ^ x $$

2. $$ (a ^ x) ^ y = a ^ (x * y) $$

3. $$ a ^ (x + y) = a ^ x * a ^ y $$

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