Question

Write the exponential equation as a logarithmic equation, or vice versa. $$ e^{-2}=0.1353 \ldots $$

Short answer

The logarithmic form of the given exponential equation \(e^{-2} = 0.1353...\) is \(\ln(0.1353...) = -2\).

Step-by-step solution

Analyze the Given Exponential Equation

The given exponential equation is \(e^{-2} = 0.1353...\). This can be interpreted as 'e' to the power of '-2' equals '0.1353...'.

Convert to a Logarithmic Equation

To convert the exponential equation to a logarithmic equation, remember that the base of the exponential equation (in this case 'e') becomes the base of the logarithm. The result of the exponential equation becomes the argument of the logarithm, and the exponent becomes the result of the logarithm. Therefore, \(e^{-2} = 0.1353...\) can be rewritten as \(\ln(0.1353...) = -2\). This is the logarithmic form of the given exponential equation.

Verify the Conversion

To verify the conversion, attempt to rewrite the logarithmic equation back into an exponential form. Using the formula \(\ln(x) = y\) as \(e^y = x\), \(\ln(0.1353...) = -2\) can be rewritten as \(e^{-2} = 0.1353...\), which matches the original exponential equation. Hence, the conversion was done correctly.

Key concepts

Exponential Equations

Understanding exponential equations is essential as they are widely used in calculations involving growth, decay, and compound interest. In an exponential equation, a constant base is raised to a variable exponent. General form of an exponential equation is expressed as a^x = b, where a is the base and is a constant, x is the exponent, and b is the result of the exponential expression.

When working with exponential equations like e^{-2} = 0.1353..., one may frequently require to solve for the exponent, which often necessitates converting the equation into a logarithmic form for ease of calculation. This process is central to managing exponential growth or decay scenarios found in real-life applications such as biology, finance, and physics.

Logarithmic Equations

A logarithmic equation involves logarithms, which are the inverses of exponential functions. The form of a logarithmic equation is \[\log_{a}(x) = c\], which is read as 'log base a of x equals c'. This implies that the base a raised to the power c equals x. They are particularly helpful when one needs to determine the exponent that a base is raised to obtain a certain number.

In the case of our exercise, e^{-2} = 0.1353... was converted to \[\ln(0.1353...) = -2\], showcasing how exponential equations can be seamlessly transformed into logarithmic ones. This transformation allows for the exponent, previously difficult to isolate, to be easily identified.

Natural Logarithm

The natural logarithm is a specific type of logarithm that is widely used due to its natural occurrence in mathematical and scientific formulas. The natural logarithm, denoted as \[\ln(x)\], uses the mathematical constant e (\textasciitilde 2.718) as its base. Its notability comes from the fact that the derivative and integral of the natural logarithm has very elegant and simple forms.

For example, \[\ln(e^{-2}) = -2\] represents the natural logarithm of the number \[0.1353...\], with the number e as the base. The equation \[\ln(x) = y\] can be switched back to its exponential form, \[e^y = x\], which is frequently used for checking if the logarithmic transformation is done correctly—as demonstrated in the step by step solution.

Properties of Logarithms

Manipulating logarithmic expressions often requires an understanding of the properties of logarithms. Some of the fundamental properties include:

• The Product Rule: \[\ln(ab) = \ln(a) + \ln(b)\]
• The Quotient Rule: \[\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\]
• The Power Rule: \[\ln(a^c) = c \cdot \ln(a)\]

These properties are invaluable when you need to simplify complex logarithmic expressions or solve logarithmic equations. For instance, when confronting an equation with a logarithm raised to a power, like \[\ln(x^2)\], one can use the Power Rule to rewrite it as \[2 \cdot \ln(x)\], often simplifying the solving process.

By employing these properties, the logarithmic terms can be adjusted to a form that makes it easier to combine, compare, or solve for the variables in question.