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Questions and Answers
What is the domain of the logarithmic function $f(x) = \log_2(x)$?
If $\log_a(x) = 2$ and $\log_b(y) = 3$, what is $\log_a(x^3y^2)$?
If $\log_2(x) = 4$, what is the value of $x$?
Which of the following is equivalent to $\log_3(9)$?
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If $\log_2(x) = y$, then $x$ is equal to:
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If $\log_a(b) = c$ and $\log_b(a) = d$, what is the relationship between $c$ and $d$?
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What is the logarithm of 1 to any base?
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If log_b(1/a)=-log_ba, then what is the expression for log_b(a^2)?
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If ln(e^5)=5, what is the value of e^(ln 5)?
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What is the value of log_10(100)=2 because _____?
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If we have a number x raised to the power of n, then what is the property expressed as?
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For any base b, what is the value of log_b(1)?
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Study Notes
Indices and Logarithms
In mathematics, indices and logarithms are closely related concepts. While indices involve raising a number to a certain power, logarithms represent the inverse operation. In particular, if a is a non-zero real number, the logarithm of a to base b, often denoted as log_ba, is the solution to the equation b^k=a. For example, log_10(1000) = 3 ("the log base 10 of 1000 is 3").
Logarithmic Functions
A logarithmic function is typically represented as f(x)=logbx, where b is the base. The domain of such a function is (0,∞) if b>1 or (-∞,0) if b<1. The range is (−∞,∞) regardless of the choice of b>1.
Laws of Logarithms
Product Rule
The product rule states that for positive numbers a and b, the logarithm of their product is the sum of the logarithms of a and b, i.e., log_ba+log_bc=log_bd. For example, if we have two numbers x and y, then ln(xy)=ln(x)+ln(y). This rule can be extended to any number of factors, as long as they are positive.
Quotient Rule
The quotient rule states that for positive numbers a and b, the logarithm of their quotient is the difference between the logarithms of a and b, i.e., log_ba−log_bc=log_bd. For example, if we have two numbers x and y, then ln(x/y)=ln(x)−ln(y). This rule also holds true for any number of terms, provided they are positive.
Power Rule
The power rule states that for a positive number a and a real exponent k, we have log_ba=klna. For example, if we have a number x raised to the power of n, then ln(x^n)=nln(x). This rule also holds true for any base.
Log of One and Log Reciprocal Rule
The log of one is always zero, regardless of the base. That is, log_be=0. Also, the log of the reciprocal of a is equal to the negative of the log of a, i.e., log_ba=−log_ba.
Common Log
Logarithms with base 10 are often referred to as common logs or decimal logs. If we write log_10x=a, it means that 10^a=x. In other words, x can be expressed as the product of powers of 10. For instance, log_10(1000)=3 because 10^3=1000. It's worth noting that the inverse operation, raising 10 to a power, is expressed as 10^a=x, where x is the original number.
Natural Log
Natural logs use Euler's constant e as the base, which is approximately equal to 2.718281828459... The natural logarithm of a number x is denoted as ln(x), and it satisfies the property that e^(ln(x))=x. Note that while natural logs do not explicitly show the base e, they implicitly assume this base when no other base is specified. For example, ln(e^x)=x because e^ln(e^x)=(e^x)^(1)=e^x=e^x.
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Description
Test your knowledge on indices and logarithms, including concepts such as logarithmic functions, laws of logarithms, common logs, and natural logs. Explore how indices and logarithms are interconnected and understand the key rules governing each concept.
