Logarithms - Theory & Concepts

Logarithms

$Learning Objectives$

Convert between exponential and logarithmic forms and explain inverse functions.
Enforce base $b>0$ , $b\neq1$ , and positive argument requirements.
Read the shape, intercepts, and asymptotes of exponential and logarithmic graphs.
Apply product, quotient, power, and change-of-base laws with their domains.
Interpret horizontal and vertical translations of logarithmic functions.
Solve exponential and logarithmic equations and reject extraneous candidates.

A logarithm is the exponent needed to obtain a positive argument from an admissible positive base. Logarithms turn multiplicative relationships into additive ones and are useful for solving for a variable appearing in an exponent.

$Definition and Domain$

For real logarithms, the base must satisfy $b>0$ and $b\ne1$ , and the argument must satisfy $x>0$ . Then $\log_b(x)=y$ if and only if $b^y=x$ . The exponential function $y=b^x$ has positive outputs; the logarithm $y=\log_b(x)$ is its inverse and thus has a positive-input domain. The logarithmic output $y$ may be any real number.

Fundamental Equivalence

Real inverse relationship with an admissible base and positive argument.

\log_b(x)=y\quad\Longleftrightarrow\quad b^y=x,\qquad b>0,\ b\ne1,\ x>0

Variables

Symbol	Description	Unit
$b$	Positive base, not equal to one	-
$x$	Strictly positive logarithm argument	-
$y$	Real logarithm value and corresponding exponent	-

$Reading the Inverse Graphs$

The pair $(0,1)$ on $y=2^x$ becomes $(1,0)$ on $y=\log_2 x$ . Reflection exchanges input and output. For $b>1$ the exponential and logarithm increase; when $0<b<1$ they both decrease. The logarithm's vertical asymptote is $x=0$ , and it has no values at or left of that boundary.

$Common and Natural Logarithms$

The common logarithm is $\log(x)=\log_{10}(x)$ ; the natural logarithm is $\ln(x)=\log_e(x)$ , where $e\approx2.71828$ . Both require $x>0$ .

$Interactive Logarithmic Functions$

Explore base values above one and strictly between zero and one. Observe how the graph reflects or changes monotonic direction without admitting nonpositive arguments.

Logarithms & Exponentials Explorer

Base (b)2

Domain: $x > 0$

Base: $b > 1$

Asymptote: The log curve approaches $x = 0$ (the y-axis) but never touches it, because $b^y \neq 0$ .

Identity: $b^{\log_b(x)} = x$

Logarithmic Function

y = \log_{2}(x)

Exponential Function

y = 2^x

Reflection Line

y = x

$Logarithm Laws and Their Conditions$

For $b>0$ , $b\ne1$ , and positive $x,y$ , the following are valid. For an arbitrary real exponent $p$ , $x^p$ is positive because $x>0$ .

Product: $\log_b(xy)=\log_b x+\log_b y$ .
Quotient: $\log_b(x/y)=\log_b x-\log_b y$ .
Power: $\log_b(x^p)=p\log_b x$ .
Change of base: $\log_b x=\dfrac{\ln x}{\ln b}=\dfrac{\log_c x}{\log_c b}$ for $c>0$ , $c\ne1$ .

These are not sum or difference rules: $\log_b(x+y)$ is generally not $\log_b x+\log_b y$ . Condensing several logarithms never removes the original argument-domain restrictions.

$Why Change of Base Works$

If $b^y=x$ , taking natural logs gives $y\ln b=\ln x$ , hence $y=\ln x/\ln b$ . Since $b\ne1$ , its logarithm is nonzero and the division is defined. Choosing common logs instead gives the same value, as the visual confirms for $x=1,2,4,8$ at base 2.

$Laws of Logarithms Simulation$

Choose a law and valid positive inputs. Compare the original expression and transformed form numerically without violating the argument conditions.

Laws of Logarithms Sandbox

Select Logarithm Law

Base (b)10

Value x10

Value y5

Expanded vs Condensed Values

Condensed Left Side

\log_{10}(10 \cdot 5)

1.6990

Expanded Right Side

\log_{10}(10) + \log_{10}(5)

1.6990

Adjust sliders to watch both condensed and expanded mathematical equations update in real-time, verifying logarithmic parity.

$Horizontal and Vertical Shifts$

For the base graph $f(x)=\log_b x$ , the shifted graph $g(x)=\log_b(x-h)+k$ moves right by $h$ and up by $k$ (negative values indicate left/down). Its domain is $x>h$ and vertical asymptote is $x=h$ . The value of $k$ changes output heights but not the domain. For example $g(x)=\log_2(x-1)+1$ is right one, up one, with domain $x>1$ .

$Check the Translation$

The base point $(1,0)$ translates to $(2,1)$ , and $(2,1)$ to $(3,2)$ . The shifted function has no real value at $x=1$ or to its left. The curves are schematic; the marked points and asymptotes are the exact checks.

$Solving Equations with Domain Checks$

In an exponential equation, isolate the power and apply a logarithm to a positive result. In a logarithmic equation, state all original positive-argument inequalities first. Condense only using identities valid on that domain, exponentiate, and substitute each candidate back into the original equation. Squaring, condensing, or multiplying by expressions can otherwise introduce inadmissible candidates.

$Solving Exponential Equations$

STEP-BY-STEP

Isolate the exponential expression so the right-hand side is positive.
Take logarithms of both positive sides.
Use the power rule to bring the exponent down.
Solve the resulting algebraic equation; check the original exponential equality.

$Solving Logarithmic Equations$

STEP-BY-STEP

Record the original argument constraints ( $>0$ ) and admissible bases.
Apply logarithm laws only where their inputs satisfy those constraints.
Isolate the logarithm, then convert to exponential form.
Solve and substitute every candidate back into the original equation.
Reject any candidate that makes an original logarithm undefined.

$Interpret a Base-Ten Logarithmic Scale$

For a positive dimensionless ratio $I/I_0$ , the idealized index $L=\log_{10}(I/I_0)$ increases by one when the ratio is multiplied by ten. A change of two corresponds to a factor of $100$ . This relationship alone is not a complete definition of earthquake magnitude, decibel sound level, or a particular scientific standard; each application has additional conventions.

$Positive Ratios Are Required$

L=\log_{10}\left(\frac{I}{I_0}\right),\quad I>0,\ I_0>0

The ratio is dimensionless and strictly positive; the base $10$ satisfies the logarithm-base restrictions.

$Exploring a Logarithmic Scale$

Use the explorer to compare additive steps in $L$ with multiplicative changes in a positive reference ratio.

Linear vs. Logarithmic Scale

Energy / Intensity Factor (

10^x

)x = 3.0

1100M

Linear Scale

Value:

1.0k

Notice how the linear bar barely moves for small values, then shoots off completely for large ones.

Logarithmic Scale (Base 10)

$\log_{10}(\text{Value}):$

3.0

The logarithmic scale grows proportionally to the exponent, compressing huge ranges into manageable numbers.

Order of Magnitude: An increase of $+1$ on a Base-10 logarithmic scale means the underlying value gets 10 times larger. An increase of $+2$ means it gets 100 times larger.

Richter Scale

A magnitude 6.0 earthquake has 10 times the wave amplitude of a 5.0.

Decibels (dB)

Every 10 dB increase represents a 10-fold increase in sound intensity.

pH Scale

A pH of 4 has 10 times the hydrogen ion concentration of pH 5.

Key Takeaways

$\log_b x=y$ means $b^y=x$ , with $b>0$ , $b\ne1$ , and $x>0$ .
Exponential and logarithmic graphs are reflections across $y=x$ ; a log graph has a vertical asymptote at the input boundary.
Product, quotient, and power laws transform multiplication, division, and powers, not addition of arguments.
Change of base divides logarithms by the logarithm of the original base.
In $\log_b(x-h)+k$ , the boundary shifts to $x=h$ and the graph translates right by $h$ and up by $k$ .
Every proposed solution of a logarithmic equation must pass the original domain conditions.

$References$

OpenStax, Precalculus 2e, §4.3–4.5: Logarithmic Functions, Graphs, and Properties. Used for domain, graph, shift, logarithm laws, and change-of-base identities; accessed October 2026.

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