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Define logarithm.
The exponent to which a base must be raised to produce a given number. If $b^a = c$, then $log_b(c) = a$.
What is the base of a common logarithm?
The base of a common logarithm is 10. It is written as $log(c)$.
What is the base of a natural logarithm?
The base of a natural logarithm is $e$ (Euler's number, ≈ 2.71828). It is written as $ln(c)$.
What is the argument of a logarithm?
The number you're taking the logarithm of.
What is the base of a logarithm?
The base is the number that is raised to a power to obtain the argument. It must be positive and not equal to 1.
Define logarithmic scale.
A scale in which units represent a multiplicative change of the base, where each unit is a power of the base.
What is the logarithm?
The exponent to which you raise the base to get the argument.
What is Euler's Number?
Euler's Number is the base of the natural logarithm, approximately equal to 2.71828.
What is the inverse function of $b^x=c$?
The inverse function is $x = log_b(c)$
What is the argument of $log_b(c)$?
The argument of $log_b(c)$ is $c$.
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, $x = 1$.
What are the differences between linear and logarithmic scales?
Linear: Units are equally spaced, fixed increment. | Logarithmic: Units represent multiplicative change, power of the base.
Compare and contrast $log(x)$ and $ln(x)$.
$log(x)$: Base 10 | $ln(x)$: Base $e$ (Euler's number).
What are the differences between exponential and logarithmic functions?
Exponential: $y = b^x$, rapid growth | Logarithmic: $y = log_b(x)$, slower growth, inverse of exponential.
Compare and contrast the graphs of $y = log_b(x)$ and $y = b^x$.
$y = log_b(x)$: Vertical asymptote at $x = 0$, passes through (1, 0) | $y = b^x$: Horizontal asymptote at $y = 0$, passes through (0, 1)
Compare and contrast the domains of $y = log_b(x)$ and $y = b^x$.
$y = log_b(x)$: Domain is $(0, \infty)$ | $y = b^x$: Domain is $(-\infty, \infty)$
Compare and contrast the ranges of $y = log_b(x)$ and $y = b^x$.
$y = log_b(x)$: Range is $(-\infty, \infty)$ | $y = b^x$: Range is $(0, \infty)$
Compare and contrast the behavior of $y = log_b(x)$ and $y = b^x$ as $x$ approaches infinity.
$y = log_b(x)$: Approaches infinity at a decreasing rate | $y = b^x$: Approaches infinity at an increasing rate
Compare and contrast the derivatives of $y = log(x)$ and $y = ln(x)$.
$y = log(x)$: Derivative is $\frac{1}{x \cdot ln(10)}$ | $y = ln(x)$: Derivative is $\frac{1}{x}$
Compare and contrast the use of linear and logarithmic scales in data representation.
Linear: Suitable for data with evenly distributed values | Logarithmic: Suitable for data with values spanning several orders of magnitude
Compare and contrast the transformations of $y = log_b(x)$ and $y = b^x$ when $b > 1$ and $0 < b < 1$.
$y = log_b(x)$: Increasing function when $b > 1$, decreasing function when $0 < b < 1$ | $y = b^x$: Increasing function when $b > 1$, decreasing function when $0 < b < 1$