Problem 80 - Calculus for Biology and Medicine

Question

Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=5^{x} ; \text { base } 5 $$

Step-by-Step Solution

Answer

Transform to linear: \(\log_5(y) = x\); plot a line with slope 1 in \((x, \log_5(y))\)-coordinates.

Step 1: Understanding Logarithmic Transformation

To transform an exponential relationship to a linear one using logarithms, we can apply a logarithmic function to both sides of the equation. In this case, we are using base 5, so we will apply the transformation using the \( \log_5 \) function.

Step 2: Apply the Logarithm on Both Sides

Start by writing the given exponential equation: \( y = 5^x\). Apply the base 5 logarithm on both sides: \( \log_5(y) = \log_5(5^x)\).

Step 3: Simplify Using Logarithm Properties

Utilize the property of logarithms that states \( \log_b(b^a) = a\). Thus, \( \log_5(5^x) = x\). So, the equation simplifies to: \( \log_5(y) = x\).

Step 4: Graph the Linear Relationship

Now that we have a linear equation \( \log_5(y) = x\), we can plot this on a set of axes. On a graph with \(x\) on the x-axis and \(\log_5(y)\) on the y-axis, you plot the line, which should be a straight line passing through the origin with a slope of 1.

Key Concepts

Exponential Relationship

An exponential relationship occurs when a variable increases rapidly by a constant factor over consistent increments. In the equation \( y = 5^x \), the variable \( y \) grows exponentially as \( x \) increases. Specifically, every time \( x \) increases by 1, \( y \) is multiplied by the base of the exponential equation, which is 5 in this case. Exponential relationships are common in scenarios like population growth, radioactive decay, and interest compounding.
To better analyze such relationships, we can transform the equation using logarithms, which simplifies the form and changes the nature of the relationship from exponential to linear, making it much easier to work with.
This transformation is key because it converts complex and fast-growing functions into simpler, predictable ones that can be analyzed using straight-line graph techniques.

Linear Graphing

Linear graphing is an essential method in studying transformed exponential relationships. After transforming the original exponential equation, like \( y = 5^x \), into a linear form \( \log_5(y) = x \), the equation can be expressed as a straight line on a graph.
This makes it far more manageable, as linear relationships are straightforward with constant rates of change.
When graphing the linear equation:

• Place \( x \) along the x-axis.
• Plot \( \log_5(y) \) on the y-axis.

The resulting graph should show a straight line through the origin, with a slope of 1.
Linear graphing of logarithmic transformations allows us to visually interpret how the exponential function is behaving in a simpler way, bringing clarity to data that would otherwise be more complex to understand.

Logarithm Properties

Logarithms possess certain properties that are instrumental in transforming exponential equations. When we apply a logarithm to each side of an equation like \( y = 5^x \), it leverages key properties to simplify the transformation. For example, the property \( \log_b(b^a) = a \) is used here.
When you encounter \( \log_5(5^x) \), it simplifies to \( x \), because the logarithm of a number base raised to an exponent returns the exponent itself.

• This simply means that applying a log base 5 to both sides peels back the exponential layer, leaving \( x \).
• Such properties are powerful in reducing complicated exponential relationships to much simpler linear ones.

Logarithms also simplify multiplications to additions, divisions to subtractions, and powers to products, making them indispensable in mathematical modelling and analysis. Understanding these properties is crucial in efficiently dealing with exponential and logarithmic functions.