What does the graph of $y = log_b(x)$ tell us about its domain?
The domain is $x > 0$, meaning the graph exists only for positive values of $x$.
What does the graph of $y = log_b(x)$ tell us about its range?
The range is all real numbers, meaning $y$ can take any real value.
How does the base $b$ affect the graph of $y = log_b(x)$?
If $b > 1$, the graph is increasing. If $0 < b < 1$, the graph is decreasing.
What does the graph of $y = log_b(x)$ tell us about its vertical asymptote?
There is a vertical asymptote at $x = 0$.
What does the graph of $y = e^x$ tell us about its derivative?
The derivative of $y = e^x$ is also $e^x$, meaning the rate of change is always positive and equal to the function's value.
What does the graph of $y = ln(x)$ tell us about its derivative?
The derivative of $y = ln(x)$ is $1/x$, which is always positive for $x > 0$.
What does the graph of $y = log(x)$ tell us about its derivative?
The derivative of $y = log(x)$ is $\frac{1}{x \cdot ln(10)}$, which is always positive for $x > 0$.
What does the graph of $y = log_b(x)$ tell us about its x-intercept?
The x-intercept is always at $x = 1$, since $log_b(1) = 0$ for any valid base $b$.
What does the graph of $y = log_b(x)$ tell us about its concavity?
The graph of $y = log_b(x)$ is concave down for $b > 1$ and concave up for $0 < b < 1$.
What does the graph of $y = log_b(x)$ tell us about its behavior as $x$ approaches infinity?
As $x$ approaches infinity, $y = log_b(x)$ also approaches infinity, but at a slower rate than a linear function.
Explain the inverse relationship between logs and exponents.
Logs 'undo' exponents. If $b^x = c$, then $x = log_b(c)$. They essentially reverse each other's operations.
Describe a linear scale.
Units are equally spaced, each representing a fixed increment.
Why are logarithmic scales useful?
They are perfect for displaying data that spans many orders of magnitude, compressing large ranges and making trends easier to see.
Explain the concept of a logarithmic scale.
A logarithmic scale represents values using the logarithm of the quantity. Equal distances on the scale represent equal ratios.
Explain the relationship between exponential and logarithmic forms.
Exponential form expresses a number as a base raised to a power, while logarithmic form expresses the power to which the base must be raised to obtain the number.
Explain why the base of a logarithm cannot be 1.
If the base were 1, $1^x$ would always be 1, regardless of the value of $x$. This would make the logarithm undefined for any number other than 1.
Explain the significance of the natural logarithm.
The natural logarithm is significant because it is based on the number $e$, which appears frequently in calculus and other areas of mathematics. It simplifies many calculations and is used in various models of natural phenomena.
Explain why the argument of a logarithm must be positive.
Because logarithms are the inverse of exponential functions, and exponential functions with real bases always produce positive values. Therefore, the logarithm of a non-positive number is undefined in the real number system.
Explain how logarithmic scales are used to compress large ranges of data.
By using logarithms, large values are transformed into smaller values, making it easier to visualize and analyze data that spans several orders of magnitude.
Describe the behavior of the graph of $y = log_b(x)$ as $x$ approaches 0.
As $x$ approaches 0, the value of $y = log_b(x)$ approaches negative infinity, assuming $b > 1$. This indicates that the function has a vertical asymptote at $x = 0$.
How to convert $2^x = 32$ to logarithmic form?
Identify the base (2), the exponent (x), and the result (32). Rewrite as $log_2(32) = x$.
How to evaluate $log_5(25)$?
Ask: 'To what power must I raise 5 to get 25?' Since $5^2 = 25$, $log_5(25) = 2$.
How to solve for $x$ in $log_3(x) = 4$?
Convert to exponential form: $3^4 = x$. Then, $x = 81$.
How to solve the equation $log_2(x) = 5$ for $x$?
Convert the logarithmic equation to exponential form: $2^5 = x$. Simplify to find $x = 32$.
How to evaluate $log(10000)$ without a calculator?
Recognize that the base is 10. Determine the power of 10 that equals 10000: $10^4 = 10000$. Therefore, $log(10000) = 4$.
How to simplify the expression $ln(e^7)$?
Use the property that $ln(e^x) = x$. Therefore, $ln(e^7) = 7$.
How to solve for $x$ in the equation $log_b(x) = y$?
Convert to exponential form: $b^y = x$.
How to solve for $x$ in the equation $2log(x) = 6$?
Divide both sides by 2: $log(x) = 3$. Convert to exponential form: $10^3 = x$. Therefore, $x = 1000$.
How to solve for $x$ in the equation $log_2(x + 1) = 3$?
Convert to exponential form: $2^3 = x + 1$. Simplify: $8 = x + 1$. Subtract 1 from both sides: $x = 7$.
How to solve for $x$ in the equation $ln(x) = 0$?
Convert to exponential form: $e^0 = x$. Since any number raised to the power of 0 is 1, $x = 1$.
