Logarithmic to Exponential Form: Convert, Evaluate, and Read the Richter Scale

Haiti: magnitude 7.0, 285,000 homes damaged. Honshu: magnitude 9.0, 332,000 buildings damaged.

The building counts sit in the same ballpark. The energies do not. The rule of the Richter scale: each whole level multiplies the energy by 10.

From 7.0 to 8.0 is one factor of 10. From 8.0 to 9.0 is another. Two factors multiply to 100 — the fact that makes logarithmic to exponential form worth learning.

So 9.0 is not "about twice" the strength of 7.0 — it is one hundred times. Reading that hidden multiplier is exactly what converting from logarithmic to exponential form trains you to do, and the chain below shows why: one look at it, and logarithmic to exponential form starts to feel real.

Set the quakes aside. Multiply 2 by itself 3 times: $2\times2\times2 = 8$.

Three numbers are on the table in this smallest logarithmic to exponential form case: base 2, count 3, result 8.

Exponential form — one half of the logarithmic to exponential form pair — states the fact forward: base 2, used 3 times, gives 8. Written $2^3 = 8$.

Logarithmic form, the other half of the logarithmic to exponential form pair, asks backward: base 2, used how many times, gives 8? The answer is still 3, written $\log_2 8 = 3$.

Same three numbers — only the roles swap, which is all a logarithmic to exponential form conversion ever is. The logarithm's answer is the exponent itself.

How many times must 2 multiply to reach 8? Count it: $2\times2\times2$ — three times.

In letters: base $a$, count $x$, result $N$.One fact, two writings — the whole logarithmic to exponential form dictionary in one line:

• Exponential form: $a^x = N$ (multiply base $a$ by itself $x$ times to get $N$)
• Logarithmic form: $x = \log_a N$ (how many times must $a$ multiply to reach $N$? — $x$)

These two sentences say the same thing. Hall and Knight's classic 1887 algebra text states it plainly: "The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number."

Plug the small example back in: $a=2$, $x=3$, $N=8$. Exponential form $2^3 = 8$; logarithmic form $\log_2 8 = 3$. Moving between them is the whole skill of logarithmic to exponential form.

Two free facts fall out of the logarithmic to exponential form relationship, true for every base:

• $\log_a 1 = 0$, because $a^0 = 1$: zero multiplications leaves 1;
• $\log_a a = 1$, because $a^1 = a$: one multiplication is the base itself.

One notation note: calculators want parentheses, as in $\log_2(8)$.

Every logarithmic to exponential form conversion takes two steps.

Step 1 — name the three roles (the logarithmic to exponential form inventory). The base never moves. In logarithmic form, the number inside the log parentheses is the result, and the number on the right side is the exponent.

Step 2 — swap sides, the actual logarithmic to exponential form move. Keep the base. The logarithm's inside number (the result) moves to the right side of the equals sign. The old right side climbs up to become the exponent.

The chant is one line: base stays, inside becomes the result, right side becomes the exponent.

Mini examples in both directions:

• $\log_3 9 = 2$ becomes $3^2 = 9$;
• $5^2 = 25$ becomes $\log_5 25 = 2$.

Read a logarithm as a question: $\log_2 32 = 5$ reads "2 to what power gives 32? — the 5th." Read enough and the logarithmic to exponential form swap turns automatic.

Problem. Rewrite each equation in exponential form, then verify it.

(a) $\log_3 9 = 2$ (b) $\log_{10} \frac{1}{10000} = -4$

Solution (a). Name the roles: base 3, result 9, exponent 2. Swap: $3^2 = 9$ — the logarithmic to exponential form steps in one line. Verify: $3\times3 = 9$. True.

Solution (b). Base 10, inside $\frac{1}{10000}$, right side $-4$. Swap: $10^{-4} = \frac{1}{10000}$.

A negative exponent takes the reciprocal — routine in logarithmic to exponential form work: $10^{-4} = \frac{1}{10^4} = \frac{1}{10000}$. True.

Answer. (a) $3^2 = 9$; (b) $10^{-4} = \frac{1}{10000}$.

Notice the instant check this example teaches: whenever the result is less than 1, the exponent must be negative. That single observation catches most slips in logarithmic to exponential form work.

Problem. Evaluate by hand — pure logarithmic to exponential form thinking, no calculator: (a) $\log_4 64$ (b) $\log_2 \frac{1}{32}$.

Solution (a). Read it as a question: 4 to what power gives 64? Walk up the ladder: $4^1 = 4$, $4^2 = 16$, $4^3 = 64$. The answer is 3.

Solution (b). 2 to what power gives $\frac{1}{32}$? First the forward direction: $2^5 = 32$. Then $\frac{1}{32} = \frac{1}{2^5} = 2^{-5}$. The answer is $-5$.

Answer. (a) 3; (b) $-5$.

Evaluating a logarithm is secretly a logarithmic to exponential form conversion: rewrite $\log_4 64$ as "4 to what power equals 64?" and the answer surfaces.

Problem. One quake releases 500 times the energy of another — the applied logarithmic to exponential form problem. The gap $x$ satisfies $10^x = 500$. Find $x$ to three decimal places.

Estimate first — the fastest logarithmic to exponential form sandwich: $10^2 = 100$, $10^3 = 1000$, and 500 sits between, so $x$ sits between 2 and 3.

Then get exact. Convert to $x = \log 500$ (the 10 is customarily left unwritten). The calculator finishes the logarithmic to exponential form conversion: $x \approx 2.699$.

Answer. $x \approx 2.699$ — the exact logarithmic to exponential form result, a gap of about 2.7 levels.

Closing the opening scene: Honshu at 9.0 stands 2.0 levels above Haiti at 7.0, and $10^{2.0} = 100$ times the energy — the same computation as this example. The number line below places every gap on one true-scale axis — the logarithmic to exponential form map of energy gaps.

1. Swapping the wrong roles. Correct: $\log_2 8 = 3$ becomes $2^3 = 8$. Wrong: $2^8 = 3$ or $8^3 = 2$ — the classic logarithmic to exponential form slip. The base never moves — the other two trade places in every logarithmic to exponential form conversion.

2. Treating the answer as the base. Correct: $\log_5 25 = 2$. Wrong: $\log_2 25 = 5$. The 2 is the answer, not the base — mixed roles sink every logarithmic to exponential form rewrite.

3. Logging a negative number. $\log_2(-8)$ does not exist. A positive base yields a positive result at every exponent, so no power of 2 ever lands on $-8$; zero is off limits the same way.

4. Misremembering $\log_a 1$. It equals 0, because $a^0 = 1$ — zero multiplications leave the number at 1. Writing 1 confuses it with $\log_a a = 1$ — both anchor quick logarithmic to exponential form checks.

5. Forgetting the base on the calculator. Pressing LOG for $\log_2 8$ returns $\log_{10} 8 \approx 0.903$. Convert bases instead: $\log_2 8 = \frac{\log 8}{\log 2} = 3$ — or just do the logarithmic to exponential form swap in your head.