**each input, x, value gives one unique output, y**, value. Each x gives only one y, and each y gives only one x. This means exponential equations

What is the one word of a person or group which attacks a place? .

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The one-to-one property of logarithmic functions tells us that, **for any real numbers x > 0, S > 0, T > 0 and any positive real number b, where b≠1 b ≠ 1** , … In other words, when a logarithmic equation has the same base on each side, the arguments must be equal.

1. loga (uv) = loga u + loga v | 1. ln (uv) = ln u + ln v |
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2. loga (u / v) = loga u – loga v | 2. ln (u / v) = ln u – ln v |

3. loga un = n loga u | 3. ln un = n ln u |

Property | Description |
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Quotient Property | aman=am−n,a≠0 |

Zero Exponent Property | a0=1,a≠0 |

Quotient to a Power Property | (ab)m=ambm,b≠0 |

Properties of Negative Exponents | a−n=1an and 1a−n=an |

- Rewrite both sides of the equation as an exponential expression with the same base. If this cannot be done, use method 2.
- Since the bases are equal, then the exponents must be equal. Set the exponents equal to each other and solve.
- Check your solution.

A function f is 1 -to- 1 **if no two elements in the domain of f correspond to the same element in the range of** f . In other words, each x in the domain has exactly one image in the range. … If no horizontal line intersects the graph of the function f in more than one point, then the function is 1 -to- 1 .

1-1 & Onto Functions. A function f from A (the domain) to B (the range) is BOTH one-to-one and onto when no element of B is the image of more than one element in A, AND all elements in B are used. Functions that are both one-to-one and onto are referred to as **bijective**.

One-to-one Functions A graph of a function can also be used to determine whether a function is one-to-one using the horizontal line test: **If each horizontal line crosses the graph of a function at no more than one point**, then the function is one-to-one.

**Not all functions have inverse functions**. The graph of inverse functions are reflections over the line y = x. … A function is said to be one-to-one if each x-value corresponds to exactly one y-value. A function f has an inverse function, f -1, if and only if f is one-to-one.

The quotient rule for logarithms says that the logarithm of **a quotient is equal to a difference of logarithms**. Just as with the product rule, we can use the inverse property to derive the quotient rule.

You can use the similarity between the properties of exponents and logarithms to find the property for the logarithm of a quotient. With exponents, to multiply two numbers with the same base, you add the exponents. To divide two numbers with the same base, you subtract the exponents.

- There are four following math logarithm formulas: ● Product Rule Law:
- loga (MN) = loga M + loga N. ● Quotient Rule Law:
- loga (M/N) = loga M – loga N. ● Power Rule Law:
- IogaMn = n Ioga M. ● Change of base Rule Law:

The Product Property of Square Roots states that the square root of a **product is equal to the product of the square roots of each of the factors**.

- All exponents in the radicand must be less than the index.
- Any exponents in the radicand can have no factors in common with the index.
- No fractions appear under a radical.
- No radicals appear in the denominator of a fraction.

Recall that the one-to-one property of exponential functions tells us that, for any real numbers b, S, and T, where b>0, b≠1 b > 0 , b ≠ 1 , **bS=bT b S = b T if** and only if S = T. In other words, when an exponential equation has the same base on each side, the exponents must be equal.

- When given a function, draw horizontal lines along with the coordinate system.
- Check if the horizontal lines can pass through two points.
- If the horizontal lines pass through only one point throughout the graph, the function is a one to one function.

The subtraction property of equality states that **you can subtract the same quantity from both sides of an equation and it will still balance**.

A one-to-one function is a function of which the answers never repeat. For example, the **function f(x) = x + 1** is a one-to-one function because it produces a different answer for every input.

An odd function is a **function f such** that, for all x in the domain of f, -f(x) = f(-x). A one-to-one function is a function f such that f(a) = f(b) implies a = b. Not all odd functions are one-to-one.

What Does It Mean if a Function Is Not One to One Function? In a function, **if a horizontal line passes through the graph of the function more than once**, then the function is not considered as one-to-one function. Also,if the equation of x on solving has more than one answer, then it is not a one to one function.

Examples on onto function Example 1: **Let A = {1, 2, 3}, B = {4, 5}** and let f = {(1, 4), (2, 5), (3, 5)}. Show that f is an surjective function from A into B. The element from A, 2 and 3 has same range 5. So f : A -> B is an onto function.

A continuous (and differentiable) function **whose derivative is always**. **positive (> 0) or always negative (< 0)** is a one-to-one function.

**If the horizontal line intersects the graph at more than one point anywhere**, then the function is not one-to-one. If the horizontal line intersects the graph at only one point everywhere, then the function is one-to-one.

A function is said to be one-to-one **if every y value has exactly one x value mapped onto it**, and many-to-one if there are y values that have more than one x value mapped onto them. This graph shows a many-to-one function.

It is possible to decide if a function is many-to-one by examining its graph. Consider the graph of y = x2 shown in Figure 13. We see that a horizontal line drawn on the graph cuts it more than once. This means that **two (or more) different inputs have yielded the same output** and so the function is many-to-one.

Let f be a function. If any horizontal line intersects the graph of f more than once, then f does not have an inverse. **If no horizontal line intersects the graph of f more than once**, then f does have an inverse. The property of having an inverse is very important in mathematics, and it has a name.

(log x)^2 is log(log x).

For all real numbers x,y, and z , if x=y , **then x+z=y+z** . For all real numbers x,y, and z , if x=y , then x−z=y−z .

What is the rule when you multiply two values with the same base together (**x2 * x3**)? The rule is that you keep the base and add the exponents. Well, remember that logarithms are exponents, and when you multiply, you’re going to add the logarithms. The log of a product is the sum of the logs.

log A + **log B = log AB**. This law tells us how to add two logarithms together. Adding log A and log B results in the logarithm of the product of A and B, that is log AB. For example, we can write. log10 5 + log10 4 = log10(5 × 4) = log10 20.

Descriptions of Logarithm Rules. Rule 1: Product Rule. The **logarithm of the product is the sum of the logarithms of the factors**. Rule 2: Quotient Rule. The logarithm of the ratio of two quantities is the logarithm of the numerator minus the logarithm of the denominator.

In the example 103 = 1000, 3 is the index or the power to which the number 10 is raised to give 1000. When you take the logarithm, to base 10, of 1000 the answer is 3. So, 103 = 1000 and **log10 (1000) = 3** express the same fact but the latter is in the language of logarithms.

- loga(xy) = logax + logay. …
- loga(x/y) = logax – logay. …
- loga(xr) = r*logax. …
- loga(1/x) = -logax. …
- logaa = 1. …
- loga1 = 0. …
- (logbx/logba) = logax.

The introduction to logarithms is placed in **intermediate algebra**.

There are four basic properties of numbers: **commutative, associative, distributive, and identity**. You should be familiar with each of these.

Quotient of powers This property states that **when dividing two powers with the same base, we subtract the exponents**.