Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-52: Approximate \(x\) to three significant figures. (a) \(\log x=3.6274\) (b) \(\log x=0.9469\) (c) \(\log x=-1.6253\) (d) \(\ln x=2.3\) (e) \(\ln x=0.05\) (f) \(\ln x=-1.6\)

Step-by-Step Solution

Answer

(a) 4260, (b) 8.90, (c) 0.237, (d) 9.97, (e) 1.05, (f) 0.202

Step 1: Convert Logarithm to Exponential

Recall that if \( \log_{10} x = a \), then \( x = 10^a \). For each part (a)-(c), apply this rule.

Step 2a: Solve for x in (a)

For \( \log x = 3.6274 \), convert to exponential form: \( x = 10^{3.6274} \). Using a calculator, \( x \approx 4260.4 \). To three significant figures, \( x = 4260 \).

Step 2b: Solve for x in (b)

For \( \log x = 0.9469 \), convert to exponential form: \( x = 10^{0.9469} \). Calculating this gives \( x \approx 8.898 \), which rounds to \( x = 8.90 \) to three significant figures.

Step 2c: Solve for x in (c)

For \( \log x = -1.6253 \), convert to exponential form: \( x = 10^{-1.6253} \). Calculating this gives \( x \approx 0.2373 \). To three significant figures, \( x = 0.237 \).

Step 3: Convert Natural Logarithm to Exponential

Recall that if \( \ln x = a \), then \( x = e^a \). For each part (d)-(f), apply this rule.

Step 3a: Solve for x in (d)

For \( \ln x = 2.3 \), convert to exponential form: \( x = e^{2.3} \). Using a calculator, \( x \approx 9.974 \), which rounds to \( x = 9.97 \) to three significant figures.

Step 3b: Solve for x in (e)

For \( \ln x = 0.05 \), convert to exponential form: \( x = e^{0.05} \). Calculating gives \( x \approx 1.051 \), which is \( x = 1.05 \) to three significant figures.

Step 3c: Solve for x in (f)

For \( \ln x = -1.6 \), convert to exponential form: \( x = e^{-1.6} \). Calculating this gives \( x \approx 0.2019 \). To three significant figures, this is \( x = 0.202 \).

Key Concepts

Exponential Functions

Exponential functions are a core concept in mathematics, particularly when dealing with logarithms. These functions have the form \(y = b^x\), where \(b\) is the base and \(x\) is the exponent.
They are incredibly powerful because they can model a wide range of phenomena in science and finance, from population growth to compound interest. A key property of exponential functions is that the rate of change of the function is proportional to the current value of the function.

In the context of converting logarithms to exponentials, if you have a logarithm such as \(\log_{10} x = a\), you can express this in exponential form as \(x = 10^a\).
This conversion is crucial for calculations in scientific contexts as it allows us to solve for \(x\) by simply using the properties of exponents. When working with natural logarithms, applying the same principle involves the constant \(e\), where if \(\ln x = a\), the exponential form is \(x = e^a\).
This is particularly useful in equations involving natural growth processes.

Significant Figures

Significant figures are important for accuracy in scientific calculations. They reflect the precision of a measurement, with each significant digit carrying meaning about the measurement's accuracy. When you express a number to a certain number of significant figures, you represent it as precisely as the data justify.

For example, to round a number to three significant figures, you must consider its digits closely. Look at all digits starting from the first non-zero digit:

• Round up if the next digit is 5 or more.
• Otherwise, keep the digit the same and ignore the rest.

Using significant figures helps communicate the level of certainty in your calculations and ensures that you're not suggesting that the results are more precise than they are. In our examples, such as rounding 4260.4 to 4260, you ensure that the calculation aligns with the precision of the initial data provided.

Natural Logarithms

Natural logarithms, or \(\ln\), are logarithms to the base of the mathematical constant \(e\), where \(e \approx 2.718\).
The natural logarithm is foundational in many areas of mathematics because the constant \(e\) often arises in natural processes, like growth and decay.

The natural logarithm tells us the power \(x\) that the base \(e\) must be raised to in order to equal a certain number.
For instance, if \(\ln x = a\), then \(x = e^a\). This transformation from logarithmic form to exponential form is vital in solving equations involving \(e\).

Natural logarithms simplify the math of exponential growth and decay, making them essential tools in fields such as biology, economics, and engineering. They are particularly useful when dealing with continuous growth processes, providing a more refined approach compared to base 10 logarithms.