Question

Solve each equation. $$ 4 e^{x+1}=5 $$

Short answer

x = \( \text{ln}\bigg(\frac{5}{4}\bigg) - 1 \)

Step-by-step solution

Isolate the exponential term

Start by isolating the exponential term on one side of the equation. Divide both sides by 4 to get: \[ e^{x+1} = \frac{5}{4} \]

Apply the natural logarithm

Take the natural logarithm (\( \text{ln}() \)) of both sides to remove the exponent. This will give: \[ \text{ln}(e^{x+1}) = \text{ln}\bigg(\frac{5}{4}\bigg) \]

Simplify using logarithmic properties

Use the property \( \text{ln}(e^y) = y \) to simplify the left side of the equation: \[ x+1 = \text{ln}\bigg(\frac{5}{4}\bigg) \]

Solve for x

Subtract 1 from both sides to isolate \( x \): \[ x = \text{ln}\bigg(\frac{5}{4}\bigg) - 1 \]

Key concepts

exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The general form is \(a^x\), where 'a' is the base and 'x' is the exponent. These functions are pivotal because they grow very quickly.
For example, in the equation \(4e^{x+1} = 5\), \(e\) (approximately 2.718) is the base and \(x+1\) is the exponent. Exponential functions are commonly found in problems involving growth and decay.
Some properties include:

• The value of an exponential function never reaches zero.
• It is always positive.
• Exponential growth means the function doubles at a consistent rate.

The steps to solve our example involve isolating the exponential part first because it lets us manipulate the equation using logarithms, which are the natural counterparts of exponents.

natural logarithm

The natural logarithm, denoted as \( \text{ln}() \), is a logarithm with base 'e'. It's an inverse operation to exponentiation with base 'e'. Applying \( \text{ln}() \) to both sides of an equation helps to 'undo' the exponentiation.
For instance, in the equation \[ e^{x+1} = \frac{5}{4} \], we apply \( \text{ln}() \) to both sides to get \[ \text{ln}(e^{x+1}) = \text{ln}\bigg( \frac{5}{4} \bigg) \].
This operation simplifies the problem efficiently.
Natural logarithms have several notable properties:

• \( \text{ln}(1) = 0 \)
• \( \text{ln}(e) = 1 \)
• \( \text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b) \)
• \( \text{ln}(a^b) = b \times \text{ln}(a) \)
• They are the inverse of exponentiation: \( \text{ln}(e^x) = x \)

In our case, applying \( \text{ln} \) let us remove the exponent so that solving for \(x\) becomes straightforward.

logarithmic properties

Logarithmic properties are essential tools for solving exponential equations. They help transform multiplicative relationships into additive ones, making equations easier to handle.
Key properties include:

• \( \text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b) \)
• \( \text{ln}( \frac{a}{b} ) = \text{ln}(a) - \text{ln}(b) \)
• \( \text{ln}(a^b) = b \times \text{ln}(a) \)
• Inverse property: \( \text{ln}(e^x) = x \)

In our example, the property \( \text{ln}(e^y) = y \) was critical. We applied it to simplify \( \text{ln}(e^{x+1}) \) to \( x + 1 \). This property essentially undoes the exponentiation, allowing us to solve for the variable \( x \) more easily.
By knowing how and when to apply these properties, solving exponential equations becomes a manageable task.