Tuesday, September 1, 2026

Reviewing Algebra II - Part Two: Exponents & Solving Exponential Equations

  Simplifying and solving exponential equations is one of the more important aspects given the wide array of applications in the physical sciences, such as radioactive decay, obtaining the half life of an isotope etc. So in this Part Two we focus on reviving the skills involved in solving these equations - which are  fundamental part of Algebra II- or learning them for the first time. Again, this is also based on the rules and properties of manipulating exponents which are generally part of Algebra I.   For example, we already saw in the previous post one of these rules applied to quotients, i.e.

a m /a n     =  a m - n

So for: 

a 5 /a 3   we get:    a 5 - 3    =  a 

Other rules for exponents:

(i)  a -n  =  1/a n

(ii)   (a m) n     =  a m n 

(iii)   a 1/n    =   n Ö a

(iv)   a m/ n     =   (a 1/n ) m     = (a m) 1/n

=  ( n Ö a ) m     =   n Ö( a m )

There are generally 2 ways to solve exponential equations: i) where both sides of the equation have the same base, and (ii) when they have different bases.

For case (i) consider:   

3 3 2x - 2

Here the same base (3) allows us to solve for x by equating the exponents:

x =  2x - 2 

so:  x = 2   which can be checked by plugging x = 2 back into the eqn.  Then:

3 =  3 2(2) - 2

3 32


When different bases are involved, we have basically 2 choices: a) work to get both sides to the same base, then proceed as shown for case (i) or b) crank out the needed value using logs, say base 10 or natural. Since most scientific work is now with the latter we use that:  For example, take the equation:  4= 15

We begin by taking the natural logarithm of both sides:

ln 4  =   ln 15   

Then use the power rule: i.e. ln b  = x ln b, to bring the variable exponent (x) to the front, i.e.

x ln 4 =  ln 15

We then can solve by dividing both sides by ln 4:

x =    ln 15/  ln 4  =   2.708/ 1.386  = 1.953

The other approach, usually easier, is to bring each side of an exponential equation to the same base. For example, say: 5x -1 = 0.04

The trick here is to get both sides in base 5 and go from there. It helps if we can see the decimal 0.04 can be expressed as a fraction, 0.04 = 4/ 100.

But 4/100 is just 1/ 25 and we immediately see  52 =  25, and we have our base 5.

Then,  working through we get:  5x -1 = 1/ 4x = 15
 =    5-2

So we can now equate the exponents:  x - 1 =  -2 or x = -2 + 1  = -1

Practice Problems

Solve each equation for the unknown and check:

 (1) 2a-3 - 1 = 15

(2) 9x =  55

3) 5x1/2 + 7   = 22

4) 3a3   = 81

5) 4x = 8 x+1

6) b-5 = 1/32
 
7)  3 +   7 x-1  =  10

For each of the following, write as a power with positive exponents in simplest form:

8)

9)

10)



Holman Jenkins Jr's Solution For 'Young Socialists' Seeking Relief: "Join The Army!"

 

 

Holman Jenkins Jr. -he whose bellowing twaddle dwarfs his actual mental capacity- again can't help himself WSJ, ('The Socialism We Already Have', Aug. 26, p. A13).  Therein, he not only conflates socialism with 'property tax breaks' (for homeowners over 65) but also "redistributing resources from young to old"- a tired trope I already demolished in two previous posts, e.g.

NY Times' Douthat (& Yale's Moyn) Wrong As Ever About "Older Americans Hoarding The Future"

And:

Yale Econ Prof Claims "Older People" in U.S. Are A "Gerontocracy Stealing From the Young" - Where and How He's Off Base

But what would you expect from an op-ed writer so Trumpian he lies like he breathes, and gets away with a lot because he's confident few will check his howlers and detritus. Like when he claimed in a Mar. 22-23 WSJ column ( 'CBS Kicks The UFO Habit', p. A13)  the Nimitz F-18 pilots were actually chasing "a 46' Chinese drone", not a genuine UFO -reported by Ret. Lt. Cmdr. David Frav0r as an incredibly high velocity craft reaching the cap point in seconds-  e.g. his House testimony.   

Or, when he denied in one nutso column from 2020 ('Disinfecting Journalistic Ethics',. WSJ, May 6, p. A13)  that Dotard ever stated people ought to consider 'disinfectant'.   Not prepared for the actual video that circulated contradicting his lie, i.e.

https://www.youtube.com/watch?v=M9kWsUwTcj8

But look, this is how Holman rolls, and he hasn't let up since actively pushing his 'Russia Hoax' baloney for the last x years. Despite the fact that former AF officer Reality Winner ended up in a federal slammer for 5 years  (under the Espionage Act) for leaking docs pertaining to  Russian interference in the 2016 United States elections.   Which, of course, Jenkins has termed "a hoax". 

In his latest  column, Jenkins continues his deformations of the truth often mixing them with half-truths to try to gain more traction among the weaker minded. He is totally convinced that if he offers enough 'loaded' malarkey combined with codswallop he can get enough WSJ readers to bite.  For example, quoting Richard Posner's book, 'Aging and Old Age' from which he unloads this chestnut: "Posner gives a frank discussion of the relationship between the short time horizon in the elderly and unchecked selfishness."

So Jenkins Jr. like Parsons commits the same logical fraud: conflating two groups of elders - one in the elite or upper 1 percent (like Steve Forbes and Warren Buffet).  The others in the lower 30 or 40 percent just scraping by with Social Security and some benefit from reduced property taxes- because of the Homestead Exemption in their states. (Btw, the reduced property taxes for those over 65 do have a rationale - because they no longer have children who need to attend schools in their districts - which property taxes pay for).

But the biggest outrage is when Holman stretches his definition of socialism to give a twisted bit of advice to the "young socialists seeking relief from having to be self supporting".  Offering this:

"For them, there's always the Army.  They'll know when their next meal is coming from and what to wear. And accommodations will be provided."

Adding: "Of course, the U..S. military prizes initiative. A lesson for the young socialists: You can't have everything."

Does this fulsome twit merit a rap on his shroom-gobbling noggin or what? 

See Also:

by Dean Baker | August 28, 2026 - 5:01am | permalink

— from Beat the Press

An extreme position does not become less extreme just because someone can put forward one that’s even more extreme. Massacring 100 children doesn’t become a moderate position just because someone is advocating killing 200 children.

This is how we should view the line being pushed by “moderate” voices that we have to deal with the $40 trillion debt with both spending cuts and tax hikes. The reality is that, apart from the military and Homeland Security, there is little fat in spending to be cut, as even Elon Musk inadvertently acknowledged. Insofar as we have a deficit problem, the issue is on the tax side, as can be easily shown. The rich have been taking an ever-larger share of national income over the last half-century, and they don’t feel like paying taxes on their winnings.

» article continues...

And:

by Chuck Collins | August 27, 2026 - 4:57am | permalink

— from OtherWords

The debate over capitalism vs. socialism often looks like a battle of cartoon caricatures: Monopoly Man versus the Marx Monster.

Across the planet there are countless flavors of capitalism and socialism, unique mashups of free markets and government policies. Into this muddle wade the right-wing ideologues who fear the red “socialist” demon hiding in every corner.

If Fox News existed in the 1950s, they would’ve labeled Republican President Dwight Eisenhower a socialist for taxing the wealthy and making broad public investments to expand homeownership, enable debt-free higher education, and build infrastructure and technology that expanded the modern middle class.

Yet, what do these labelers call modern-day China, a country that mints hundreds of billionaires a year and dominates global markets with private production? Are they socialist or capitalist?

» article continues...

And:

by Robert Reich | May 1, 2026 - 5:03am | permalink

— from Robert Reich's Substack

`
David Zaslav, CEO of Warner Bros. Discovery, has reason to smile

Friends,

Warner Bros. Discovery shareholders voted last Thursday on the Ellison family’s purchase of the company. Some 1.743 billion shares were cast in favor of the sale; 16.3 million were cast against it, a ratio of roughly 99 to 1.

1. Great for a Handful of Super-Wealthy, but Bad for Workers and Bad for America

This vote came soon after more than 4,000 workers in the media industry — directors, screenwriters, producers, actors, editors, cinematographers, musicians, and composers — signed a letter predicting an industry disaster if the sale went through.

» article continues...

And:

Youtube videos:


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