Rearranging Biochemical Equations with Scientific Notation and Logarithms

Welcome. This first lesson establishes a quantitative language used throughout biochemistry: powers of ten, scientific notation, and base-10 logarithms. These tools let us work comfortably with concentrations, equilibrium constants, and other quantities that can differ by many orders of magnitude.

By the end, you should be able to express biochemical quantities in scientific notation, perform the exponent arithmetic needed to rearrange simple equations, and undo a base-10 logarithm to isolate an unknown. These skills will support later work with molarity, pH, buffers, equilibrium, and enzyme kinetics.

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Scientific notation: keeping scale visible

A biochemical concentration such as $0.000072\mathrm{M}$ is awkward to read and easy to mistype. Scientific notation separates a number into:

$$c\times 10^n$$

where $c$ is a coefficient between 1 and 10, and $n$ is an integer exponent.

For example:

$$0.000072\mathrm{M}=7.2\times 10^{-5}\mathrm{M}$$

The negative exponent says that the decimal point must move five places to the left from $7.2$. Conversely:

$$3.6\times 10^4=36,000$$

A quick sense check is useful:

• A positive exponent usually produces a number greater than 1.
• A negative exponent usually produces a number between 0 and 1.
• The coefficient in normalized scientific notation must satisfy

$$1\le c<10$$

For instance, $42.8\times 10^{-6}$ has the right value but is not normalized. Moving the decimal one place left makes the coefficient $4.28$, so the exponent must increase by one:

$$42.8\times 10^{-6}=4.28\times 10^{-5}$$

This adjustment preserves the value.

Operations you will use most

When multiplying quantities in scientific notation, multiply the coefficients and add the powers of ten:

$$(6.0\times 10^{-6})(2.5\times 10^3)$$ $$=(6.0)(2.5)\times 10^{-6+3}$$ $$=15.0\times 10^{-3}$$ $$=1.50\times 10^{-2}$$

When dividing, divide the coefficients and subtract the denominator exponent:

$$\frac{8.4\times 10^{-4}}{2.1\times 10^{-6}}$$ $$=\left(\frac{8.4}{2.1}\right)\times 10^{-4-(-6)}$$ $$=4.0\times 10^2$$

Notice the critical sign step:

$$-4-(-6)=-4+6=2$$

This is particularly important in biochemistry because concentrations and constants frequently have negative exponents.

Addition and subtraction work differently. You may combine coefficients only after the exponents match:

$$3.2\times 10^{-4}+5.0\times 10^{-5}$$

Rewrite the second term with an exponent of $-4$:

$$5.0\times 10^{-5}=0.50\times 10^{-4}$$

Then add:

$$3.2\times 10^{-4}+0.50\times 10^{-4}=3.70\times 10^{-4}$$

Do not add exponents when adding quantities. The rule “add exponents” belongs only to multiplication of powers with the same base.

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Rearranging biochemical equations with powers of ten

Rearranging an equation means isolating the unknown while preserving equality. Every operation performed on one side must also be performed on the other.

Consider the acid-dissociation expression:

$$K_a=\frac{[H_3{O}^{+}][A^{-}]}{[HA]}$$

Suppose you need to isolate the hydronium concentration. First multiply both sides by $[HA]$:

$$K_a[HA]=[H_3{O}^{+}][A^{-}]$$

Then divide both sides by $[A^{-}]$:

$$[H_3{O}^{+}]=\frac{K_a[HA]}{[A^{-}]}$$

The algebra comes first. Only after the unknown is isolated should you substitute numbers.

For example, let:

$$K_a=1.8\times 10^{-5}$$ $$[HA]=0.20\mathrm{M}$$ $$[A^{-}]=0.050\mathrm{M}$$

Then:

$$[H_3{O}^{+}]=\frac{(1.8\times 10^{-5})(0.20)}{0.050}$$

Calculate the ordinary numerical ratio:

$$\frac{(1.8)(0.20)}{0.050}=7.2$$

so:

$$[H_3{O}^{+}]=7.2\times 10^{-5}\mathrm{M}$$

The scientific-notation exponent remained $-5$ because the other terms were written as ordinary decimals. If all terms are in scientific notation, apply the multiplication and division rules explicitly.

A productive routine is:

1. Write the symbolic rearrangement.
2. Substitute values, including units.
3. Separate coefficient arithmetic from exponent arithmetic.
4. Normalize the final result so the coefficient is between 1 and 10.
5. Check scale and units. A concentration should not emerge with units of ${\mathrm{M}}^2$ or ${\mathrm{M}}^{-1}$ unless the equation specifically calls for them.

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Logarithms: exponents written in reverse

A base-10 logarithm answers one question:

To what power must 10 be raised to obtain this number?

For example:

$$10^3=1000$$

is equivalent to:

$$\log (1000)=3$$

Likewise:

$$10^{-4}=0.0001$$

is equivalent to:

$$\log (0.0001)=-4$$

In chemistry, $\log$ without a written base conventionally means ${\log }_{10}$, the base-10 or common logarithm. Do not confuse it with $\ln$, which is the natural logarithm with base $e$. Both occur in chemistry, but this lesson focuses on base 10.

A useful identity for scientific notation is:

$$\log (c\times 10^n)=\log (c)+n$$

For a concentration of $2.8\times 10^{-4}$, for instance:

$$\log (2.8\times 10^{-4})=\log (2.8)-4$$

Since $\log (2.8)$ is approximately $0.447$,

$$\log (2.8\times 10^{-4})\approx -3.553$$

The log compresses a large range of concentrations into a manageable numerical scale. A tenfold increase in a quantity changes its base-10 log by exactly 1.

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The p-function and its inverse

Many chemical scales use a “p-function,” defined generally as:

$$pX=-\log (X)$$

The most familiar case is pH:

$$\mathrm{pH}=-\log [H_3{O}^{+}]$$

At the introductory level, $[H_3{O}^{+}]$ is treated as the hydronium concentration in $\mathrm{M}$. The minus sign matters: because most aqueous hydronium concentrations are less than 1, their logs are negative, and pH becomes positive.

For the concentration used above:

$$[H_3{O}^{+}]=2.8\times 10^{-4}\mathrm{M}$$ $$\mathrm{pH}=-\log (2.8\times 10^{-4})$$ $$\mathrm{pH}=-(-3.553)$$ $$\mathrm{pH}=3.55$$

For now, treat this as an illustration of logarithmic notation rather than a full pH-calculation lesson. You will return to strong acids, strong bases, pOH, and the relation between pH and pOH later in this module.

The more important algebraic move is reversing the calculation. Start from the generic p-function:

$$pX=-\log (X)$$

Multiply both sides by $-1$:

$$-pX=\log (X)$$

Now raise 10 to the power of both sides:

$$10^{-pX}=10^{\log (X)}$$

Because exponentiation with base 10 and a base-10 logarithm are inverse operations:

$$X=10^{-pX}$$

For pH, this becomes:

$$[H_3{O}^{+}]=10^{-\mathrm{pH}}$$

That is a rearrangement, not a new fact to memorize separately. Remember: the inverse of $\log$ is raising 10 to a power.

A generic logarithmic rearrangement

The same pattern appears in many biochemical equations. Suppose:

$$y=a+\log (x)$$

To solve for $x$, first isolate the logarithm:

$$y-a=\log (x)$$

Then take 10 to the power of both sides:

$$x=10^{y-a}$$

A later example is the Henderson–Hasselbalch equation:

$$\mathrm{pH}=p{K}_a+\log \left(\frac{[A^{-}]}{[HA]}\right)$$

Without yet calculating a buffer composition, you can rearrange it correctly. Subtract $p{K}_a$ from both sides:

$$\mathrm{pH}-p{K}_a=\log \left(\frac{[A^{-}]}{[HA]}\right)$$

Then apply the base-10 inverse:

$$\frac{[A^{-}]}{[HA]}=10^{\mathrm{pH}-p{K}_a}$$

The ratio is dimensionless because both numerator and denominator are concentrations in the same units. This is one reason logarithms fit naturally into biochemical equations: their arguments must be pure numbers or dimensionless ratios.

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Calculator habits and error checks

Most scientific calculators accept scientific notation through a key labeled EE, EXP, or ×10^x. Enter $2.8\times 10^{-4}$ as $2.8$, then the exponent key, then $-4$. Do not type an ordinary multiplication sign before using the EE or EXP key; that key already represents the power-of-ten part.

For logarithms:

• Use LOG for base-10 logarithms.
• Use the inverse LOG or $10^x$ function to calculate an antilog.
• Use LN only when an equation explicitly contains $\ln$.

Two checks catch many mistakes:

1. Log-domain check: You cannot take $\log (0)$ or the log of a negative number. A negative concentration or zero concentration in this context signals an earlier error.
2. Magnitude check: If a concentration is $10^{-3}\mathrm{M}$, its p-value should be near 3, not $-3$. The negative sign in a p-function reverses the sign of the logarithm.

When reporting results, retain units for concentrations and omit units for logarithms, pH, pOH, and ratios. In later quantitative work, you will also apply significant-figure rules: the number of decimal places in a pH value is tied to the number of significant figures in the concentration used to calculate it.

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Key takeaways

Scientific notation makes biochemical scales manageable:

$$c\times 10^n$$

When multiplying, add exponents; when dividing, subtract exponents. In an equation, isolate the unknown symbolically before substituting numerical values.

A base-10 logarithm and a power of 10 are inverse operations:

$$\log (x)=y$$ $$x=10^y$$

For p-functions, the leading negative sign gives:

$$pX=-\log (X)$$ $$X=10^{-pX}$$

Next, you will apply this numerical fluency to a foundational laboratory skill: calculating molarity and performing single-step solution dilutions.